DIRECT  AND  ALTERNATING 
CURRENT  TESTING  > 


BY 


FREDERICK   BEDELL,  PH.D. 
n 

PROFESSOR    OF   APPLIED    ELECTRICITY    IN   CORNELL   UNIVERSITY 


ASSISTED   BY 

CLARENCE   A.  PIERCE,  PH.D. 

SECOA^D  IMPRESSION 


Of    THE 

UNIVERSITY 

OF 


NEW    YORK 
D.    VAN    NOSTRAND   COMPANY 

23  MURRAY  AND  27   WARREN  STREETS 

LONDON :    CONSTABLE  &  CO.,  LTD. 

1909 


Copyright,  1909 
By  FREDERICK  BEDELL 

All  rights  reserved 


Printed  September,  1909 
Reprinted  November.  1909. 


PRESS  OF 
LANCASTER.  PA. 


PREFACE. 


This  manual  consists  of  a  series  of  tests  on  direct  and  alter- 
nating current  apparatus,  selected  with  reference  to  their  practical 
usefulness  and  instructive  value.  While  the  book  has  been  pre- 
pared primarily  for  students,  it  is  hoped  that  it  may  prove  helpful 
to  others.  The  presentation  is  in  the  form  of  a  laboratory 
manual;  the  author,  however,  has  not  restricted  himself  to  a 
mere  statement  of  instructions  for  conducting  tests  but  has 
directed  the  reader's  attention  to  the  principles  that  underlie  the 
various  experiments  and  to  the  significance  of  the  results.  Ex- 
perience has  shown  that  theory  is  more  readily  grasped  when  it  is 
thus  combined  with  its  application  and  that  the  application  is 
more  intelligently  made  when  its  broader  bearings  are  understood. 
The  material  has  been  systematically  arranged  and  it  is  believed 
that  the  book  may  be  found  useful  for  reference  or  as  a  text, 
aside  from  its  use  in  testing. 

From  the  text  proper  are  excluded  specialized  tests  and  those 
that  are  of  limited  application  or  require  unusual  testing  facili- 
ties, such  tests  being  described  in  the  appendices  to  the  several 
experiments.  These  appendices  thus  permit  a  fuller  discussion 
of  some  of  the  details  of  the  tests  and  various  modifications  than 
could  be  included  to  advantage  in  the  text  proper.  The  tests  in 
general  are  those  that  can  be  performed  in"  any  college  laboratory. 

No  attempt  has  been  made  to  make  the  work  exhaustive  or 
complete ;  on  the  contrary  every  effort  has  been  made  to  eliminate 
matter  of  secondary  importance  and  that  which  is  of  questionable 
technical  or  pedagogical  value. 

The  aim  has  been  to  arrange  an  introductory  series  of  experi- 
ments of  a  comprehensive  nature,  so  that  in  a  reasonable  time 
and  with  a  reasonable  amount  of  effort  the  student  may  acquire 


vi  PREFACE. 

the  power  to  proceed  to  problems  requiring  a  continually  increas- 
ing initiative  and  originality.  Although  standardized  tests  afford 
the  quickest  way  for  obtaining  certain  desired  results  and,  in  th£ 
case  of  a  student,  for  obtaining  a  knowledge  of  testing  methods, 
the  ability  to  conduct  such  tests  with  full  instructions  given  is 
soon  acquired.  Beyond  this  point  the  exclusive  use  of  standard- 
ized tests  should  be  avoided.  Standards  in  electricity  serve  best 
as  new  points  of  departure.  The  student  who  is  to  become  more 
than  the  "  ordinary  slide-rule  engineer  "  or  "  mental  mechanic  " 
will  have  sufficient  intellectual  curiosity  to  desire  more  than  any 
standardized  tests  can  give  him  and  should  be  encouraged  in  every 
way  to  seek  new  results  and  to  devise  ways  and  means  for  obtain- 
ing them  with  the  facilities  at  hand.  To  attempt  to  formulate 
such  work  would  at  once  deprive  it  of  its  freshness.  The  student 
may  well  be  referred  to  the  current  technical  press  and  to  the 
transactions  of  the  engineering  societies  for  suggestions  as  to 
subject  matter  for  further  study  and  also  as  to  methods  to  be 
adopted. 

With  reference  to  prepared  blanks  and  forms,  the  writer  be- 
lieves that  their  use  can  be,  and  often  is,  carried  too  far,  leading 
perhaps  to  good  technical  but  not  to  good  pedagogical  results.  In 
a  certain  sense  the  one  who  prepares  the  forms  and  lays  out  the 
work  is  the  one  who  really  performs  the  experiment,  the  tabu- 
lators of  data  being  assistants  who,  for  commercial  work,  require 
only  a  common  school  education. 

Progress  undoubtedly  results  from  the  development  of  indi- 
vidualism and  if  room  for  such  development  is  to  be  given  in  a 
college  course — specifically  in  a  college  laboratory  course — the 
more  or  less  standardized  instruction  must  needs  be  reduced  so 
as  not  to  fill  the  entire  available  time.  The  natural  tendency  has 
been  quite  the  reverse.  Two  decades  ago,  the  study  of  electrical 
engineering  meant,  practically,  the  study  of  direct  currents,  there 
being  little  else.  Laboratory  courses  were  developed  in  which  the 
whole  available  time  was  well  filled  with  test  after  test  upon 


PREFACE.  vii 

direct  current  generators  and  motors.  The  transformer  and 
alternator  were  then  added,  with  extensive  time-consuming  tests, 
with  the  apparent  assumption  that  the  full  development  of  alter- 
nating currents  was  reached.  In  succeeding  years  came  the  gen- 
eral development  of  polyphase  currents,  the  rotary  converter,  in- 
duction motor,  etc.,  these  subjects  being  added  to  a  crowded 
course  by  a  process  of  compression  rather  than  judicious  elimina- 
tion. The  student  was  given  more  than  he  could  possibly  assimi- 
late. As  types  of  machines  have  multiplied,  it  would  take  years 
to  perform  all  the  permutations  and  combinations  of  tests  on  all 
the  different  types.  But  is  this  necessary  for  a  student?  Why 
not  develop  a  student's  powers  by  a  few  typical  experiments  on  a 
few  typical  kinds  of  apparatus? 

With  this  end  in  view  the  writer  has  made  selection  from  ma- 
terial which  has  long  been  collecting  in  the  form  of  typewritten 
outlines.  These  have  been  in  a  process  of  continual  evolution, 
frequently  rewritten  and  used  by  many  classes.  By  a  process  of 
elimination  and  survival,  experiments  consisting  of  a  large  amount 
of  mechanical  data-taking  and  tabulation  and  a  relatively  small 
amount  of  technical  content  have  been  dropped  in  favor  of  those 
experiments  which  have  proved  most  effective  in  student  de- 
velopment. Various  demands  upon  the  writer's  time  prevented 
his  preparing  for  the  press  a  book  on  testing  a  number  of  years 
ago  and  the  present  appearance  of  the  book  is  due  in  no  small 
way  to  the  valuable  assistance  of  Dr.  Pierce.  Meanwhile  several 
admirable  manuals  have  appeared,  which  differ,  however,  in  aims 
or  scope  from  the  present  work.  The  author  hopes  to  find 
leisure,  in  the  near  future,  to  make  good  some  of  the  omissions  of 
the  present  volume  and  to  include  in  a  later  edition  additional 
chapters  on  alternating  current  motors  and  converters. 

The  present  work  is  self-contained  and  requires  only  such  pre- 
liminary courses  in  physical  and  electrical  measurements  as  are 
usually  given  in  colleges.  The  book  may  be  used  to  advantage  in 
conjunction  with  standard  texts  on  electrical  engineering,  as  those 


viii  PREFACE. 

by  Franklin  and  Esty,  S.  P.  Thompson,  and  Samuel  Sheldon,  or 
with  an  introductory  text  such  as  that  by  H.  H.  Norris.  The 
experiments  given  in  the  book  may  be  supplemented  by  others  of 
an  elementary,  intermediate  or  advanced  nature,  as  circumstances 
may  require.  The  division  of  experiments  into  parts  and  sections 
will  be  found  to  add  materially  to  the  flexibility  of  the  book. 

The  author  desires  to  express  his  appreciation  of  the  initial 
instruction  and  inspiration  of  Professor  H.  J.  Ryan  and  of  the 
continuous  cooperation  for  many  years  of  Professor  G.  S.  Moler. 
He  wishes  also  to  express  his  indebtedness  to  many  who  have  been 
associated  with  him  in  laboratory  instruction,  in  particular  to  Dr. 
A.  S.  McAllister,  as  many  references  in  the  present  text  bear  evi- 
dence. He  likewise  desires  to  express  his  appreciation  of  the 
spirit  of  cooperation  shown  by  Professors  H.  H.  Norris  and  V. 
Karapetoff  and  other  engineering  colleagues.  The  author  is  in- 
debted, furthermore,  to  various  professors  and  students,  who 
have  used  and  corrected  this  book  in  proof  during  the  last  year 
and  to  a  number  of  engineers  who  have  looked  over  the  proof 
sheets  and  have  made  valuable  suggestions.  For  all  shortcomings 
the  author  alone  is  responsible. 

ITHACA,  N.  Y.,  July  i,  1909. 


CONTENTS. 


CHAPTER   I. 

DIRECT  CURRENT  GENERATORS. 

PAGE. 
EXPERIMENT   i-A.     Generator   Study   and   Characteristics   of   a 

Series  Generator  i 

EXPERIMENT  i-B.     Characteristics  of  a  Compound  Generator. .     13 


CHAPTER    IT. 
DIRECT  CURRENT  MOTORS. 
EXPERIMENT  2-A.     Operation   and   Speed   Characteristics   of  a 

Direct  Current  Motor  (Shunt,  Compound  and  Differential).     27 
EXPERIMENT  2-B.     Efficiency   of  a   Direct   Current  Motor    (or 

Generator)  by  the  Measurement  of  Losses 41 


CHAPTER   III. 
SYNCHRONOUS  ALTERNATORS. 

EXPERIMENT  3~A.     Alternator  Characteristics 62 

EXPERIMENT  3~B.     Predetermination   of   Alternator   Character- 
istics       73 


CHAPTER   IV. 
SINGLE-PHASE  CURRENTS. 

EXPERIMENT  4-A.     Study  of  Series  and  Parallel  Circuits  Con- 
taining Resistance  and  Reactance 102 

EXPERIMENT  4-B.     Circle  Diagram  for  a  Circuit  with  Resistance 

and  Reactance 123 


x  CONTENTS. 

PAGE. 

CHAPTER   V. 
TRANSFORMERS. 
EXPERIMENT    5-A.     Preliminary    Study    and    Operation    of    a 

Transformer   128 

EXPERIMENT  5~B.     Transformer  Test  by  the  Method  of  Losses.    150 
EXPERIMENT  5~C.     Circle    Diagram    for    a    Constant    Potential 

Transformer 179 

CHAPTER   VI. 
POLYPHASE  CURRENTS. 

EXPERIMENT  6-A.     A  General  Study  of  Polyphase  Currents....   196 
EXPERIMENT  6-B.     Measurement  of   Power  and  Power   Factor 

in  Polyphase  Circuits 222 

CHAPTER   VII. 

PHASE  CHANGERS,  POTENTIAL  REGULATORS,  ETC. 

EXPERIMENT  7~A.     Polyphase  Transformation 241 

EXPERIMENT  7-6.     Induction  Regulators 250 


w\>*'  * 
UNIVERSltV 


OF 

ILIFORj 


CHAPTER  I. 
DIRECT  CURRENT  GENERATORS. 

EXPERIMENT  i-A.  Generator  Study  and  Characteristics  of  a 
Series*  Generator. 

PART  I.    GENERATOR  STUDY. 

§  I.  Faraday  discovered  (1831)  that  when  a  conductor  cuts 
lines  of  force  an  electromotive  force  is  generated  in  the  conductor 
proportional  to  the  rate  at  which  lines  are  cut,  and  all  dynamos 
(or  generators  as  they  are  now  commonly  called)  operate  on  this 
principle.  To  generate  an  electromotive  force,  it  is  essential  there- 
fore to  have  a  conductor  (or  several  conductors  combined  by 
various  winding  schemes)  forming  the  armature  as  one  member; 
and  to  have  lines  of  force  or  magnetic  flux  set  up  by  field  mag- 
nets which  form  the  second  member.  For  operation  it  is  neces- 
sary also  to  have  a  source  of  mechanical  powerf  by  which  either 
one  of  these  members  can  be  given  a  motion  with  respect  to  the 
other.  A  generator  may  havej  a  stationary  field  and  revolving 
armature;  or,  a  revolving  field  and  stationary  armature,  desig- 
nated as  the  revolving  field  type.  Although  the  latter  is  useful 
for  large  alternators,  serious  objections  to  it  have  been  found 
in  direct  current  machines,  for  with  the  commutator  stationary, 
the  brushes  must  revolve,  which  leads  to  difficulties  in  construc- 
tion and  operation.  It  is  the  custom,  therefore,  to  build  all 

*  Where  a  series  generator  is  not  available,  this  study  may  be  taken 
without  experimental  work  or  in  connection  with  Part  I.  of  Exp.  i-B. 

f  Power  is  required  to  overcome  friction  and  other  losses,  and  to  over- 
come a  counter  torque  (§§  1-3,  Exp.  2-A)  which  varies  with  the  load. 

t  Outside  of  this   classification   is   the   inductor  alternator  which   has   a 
stationary  armature,  a  stationary  field  winding  and  a  revolving  inductor  of 
iron;  its  study  should  be  taken  up  later  under  alternators. 
2  i 


2  DIRECT   CURRENT   GENERATORS.  [Exp. 

direct  current  generators  and  motors  with  a  stationary  field  and 
revolving  armature. 

§  2.  The  student  should  consult  any  of  many  excellent  treatises 
for  a  detailed  discussion  of  different  types  of  generators  and  if 
possible  should  note  one  or  two  machines  in  the  laboratory  or 
elsewhere  which  are  examples  of  each  important  type.  Machines 
noted  should  illustrate  the  following  terms,  some  of  which  are 
briefly  explained  in  later  paragraphs:  Stationary  field,  revolving 
field,  bipolar,  multipolar,  separately-excited,  self-excited,  series 
wound,  shunt  wound,  compound  wound,  magneto-generator,  open 
and  closed  coil  armature,  drum  armature,  Gramme  (or  ring) 
armature. 

Only  the  general  structure  of  the  various  machines  need  be 
noted.  Observe  particularly  the  magnetic  circuit  of  each  machine 
and  the  disposition  of  the  field  winding.  Keep  in  mind  that 
magnetic  flux  is  proportional  to  magnetomotive  force  (field 
ampere-turns)  divided  by  the  reluctance  of  the  complete  magnetic 
circuit,  i.  e.,  the  sum  of  the  reluctance  of  each  part  (air  gap,  core, 
etc.).  As  the  reluctance  of  any  part  of  a  magnetic  circuit  is  equal 
to  the  length  divided  by  the  product  of  the  cross-section  and  per- 
meability, it  is  obvious  that  an  unnecessarily  long  magnetic  circuit 
should  be  avoided,  a  fact  neglected  in  some  early  machines. 

§  3.  In  a  bipolar  generator,  one  pole  is  north  and  the  other 
south;  in  a  multipolar  generator  (with  4,  6,  8,  etc.,  poles)  the 
poles  are  alternately  north  and  south. 

Each  armature  conductor  accordingly  passes  first  underneath  a 
north  and  then  underneath  a  south  pole  and  has  induced  in  it 
an  electromotive  force  first  in  one  direction*  and  then  in  the 

*(§3a).  An  exception  is  the  so-called  unipolar,  homopolar  or  acyclic 
dynamo,  which  has  a  unidirectional  electromotive  force  generated  in  the 
armature  conductor;  it  accordingly  delivers  direct  current  to  the  line  with- 
out commutation.  Faraday's  disk  dynamo  (one  of  the  earliest  dynamos) 
was  of  this  type.  For  years  it  was  the  dream  of  zealous  electricians  to 
make  this  type  of  machine  practicable,  but  it  was  considered  only  as  an  in- 
teresting freak,  for  at  ordinary  speeds  the  voltage  generated  is  too  low 


i-A]  SERIES  GENERATOR.  3 

other,  i.  e.,  an  alternating  electromotive  force.  The  simplest  form 
of  generator  is  therefore  the  alternator,  the  current  being  taken 
from  the  armature  to  the  line  without  any  commutation.  If  the 
armature  is  stationary,  the  alternating  current  from  the  armature 
is  taken  directly  to  the  line ;  if  the  armature  is  revolving,  the 
armature  windings  are  connected  to  collector  rings  (or  slip  rings) 
from  which  the  current  is  taken  to  the  line  by  means  of  brushes. 

In  a  direct-current  generator  the  armature  windings  are  con- 
nected* to  the  several  segments  or  bars  of  a  commutator,  from 
which  the  current  is  taken  by  brushes  to  the  line.  The  alter- 
nating electromotive  force  generated  in  each  coil  is  thus  corn- 
mutated,  or  reversed  in  its  connection  to  the  line,  at  or  near  the 
time  of  zero  value  of  the  electromotive  force  of  the  coil. 

The  electromotive  force  in  each  coil  increases  from  zero  to 
a  maximum  and  back  to  zero,  and  at  any  instant  the  electromotive 
forces  in  the  various  individual  coils  have  different  values  rang- 
ing from  zero  to  a  maximum,  according  to  the  positions  of  the 
coils.  The  sum  of  these  coil-voltages,  as  impressed  upon  the 
line  as  terminal  voltage,  is  however  practically  constant. 

for  most  purposes.  But  changed  conditions  have  made  it  a  practical  and 
important  machine  (i)  driven  at  high  speed  by  the  steam  turbine,  or  (2) 
driven  at  moderate  speed  to  generate  large  currents  at  low  voltage  for  elec- 
trochemical work.  Dynamos  of  this  class  are  not  included  in  this  study. 
For  further  information,  see  "  Acyclic  Homopolar  Dynamos,"  by  Noeg- 
gerath,  A.  I.  E.  E.,  Jan.,  1905 ;  also,  Standard  Handbook,  or  Franklin  and 
Esty's  Electrical  Engineering.  For  description  of  some  structural  im- 
provements, see  pp.  560  and  574,  Electrical  World,  Sept.  12,  1908. 

*  (§3b).  The  details  of  armature  windings  will  not  be  here  discussed; 
they  are  amply  treated  in  many  text  and  handbooks.  In  almost  all 
machines  a  closed  coil  winding  is  used.  (The  Brush  and  T-H  arc 
dynamos  and  a  few  special  machines  use  open  coil  winding.)  In  a  closed 
winding,  the  armature  coils  are  connected  in  series  and  the  ends  closed. 
There  are  two  ways  of  connecting  the  coils  in  series:  wave  winding  and 
lap  winding.  In  the  wave  or  series  winding  there  are  always  two  brushes 
and  two  paths  for  the  current  from  brush  to  brush,  irrespective  of  the 
number  of  poles.  In  the  lap  or  parallel  winding,  generally  used  in  large 
generators,  there  are  as  many  paths  (and  brushes)  as  poles.  The  two 
schemes  are  essentially  the  same  in  a  bipolar  machine. 


4  DIRECT   CURRENT   GENERATORS.  [Exp. 

§  4.  Field  magnets  are  usually*  energized  by  direct  current 
passed  through  the  field  windings ;  permanent  magnets  being  used 
only  in  small  machines,  called  magneto-generators,  used  for  bell- 
ringers,  etc.  A  generator  is  separately-excited  or  self-excited 
according  to  whether  the  current  for  the  field  is  supplied  by  an 
outside  source  or  by  the  machine  itself.  Alternators  are  sepa- 
rately excited;  direct  current  generators  are  usually  self -excited. 

§  5.  A  direct  current  machine  (either  generator  or  motor)  may 
be:  (i)  Series  wound,  with  the  field  winding  of  coarse  wire  in 
series  with  the  armature  and  carrying  the  whole  armature  current ; 

(2)  Shunt  wound,  with  a  field  winding  of  fine  wire  in  shunt  with 
the  armature  and  carrying  only  a  small  part  of  the  whole  current ; 

(3)  Compound  wound,  with  two  field  windings,  the  principal  one 
in  shunt  and  an  auxiliary  one  in  series  with  the  armature. 

The  compound  generator  is  in  most  general  use,  being  best 
suited  for  all  kinds  of  constant  potential  service,  both  power  and 
lighting;  the  shunt  generator  performs  similar  service  but  not 
so  well.  The  characteristics  of  these  machines  will  be  studied 
fully  in  Exp.  i-B. 

The  series  generator  is  of  interest  because:  (i)  It  is  one  of 
the  earliest  types  and  of  historical  importance;  (2)  It  is  the 
simplest  type  and  illustrates  in  a  simple  manner  the  principles 
which  underlie  all  dynamo-electric  machinery,  both  generators 
and  motors;  (3)  In  a  compound  wound  "generator  or  motor,  the 
series  winding  is  an  important  factor  in  the  regulation  of  poten- 
tial or  speed.  In  itself  the  series  generator  is  of  relatively  small 
importance,  because  neither  current  or  voltage  stay  constant;  it 
is  used  only  in  some  forms  of  arc  light  machines  with  regulating 
devices  for  constant  current. 

In  direct  current  motors,  all  three  types  of  winding  are  em- 
ployed :  series  wound  motors  for  variable  speed  service  in  traction, 
crane  work,  etc.;  shunt  and  compound  wound  (including  differ- 

*The  induction  generator,  to  be  studied  in  a  later  experiment,  does  not 
come  under  this  classification. 


i-A]  SERIES  GENERATOR.  5 

ential  wound)   motors   for  more  or  less  constant  speed  service 
(Exp.  2-A). 

PART  II.  CHARACTERISTICS  OF  A  SERIES  GENERATOR. 

The  characteristic  curves  to  be  obtained  are:  the  magnetisa- 
tion curve,  with  the  machine  separately  excited;  the  external 
series  characteristic,  with  the  machine  self  excited;  and  the  total 
characteristic,  which  is  computed. 

§  6.  Magnetization  Curve. — This  curve  shows  the  armature 
voltage  (on  open  circuit)  corresponding  to  different  field  currents 
when  the  generator  is  separately  excited  from  an  outside  source, 
as  in  Fig.  I.  No  load  is  put  upon  the  machine.  Means  for  vary- 
ing the  field  current  must  be  provided ;  see  Appendix,  §  14. 

§  7.  Data. — Readings  are 
taken  of  field  current,  ar-        ^  ^  A 

mature  voltage  and  speed ; 
the  first  reading  is  taken 
with  field  current  zero, 

showing  the  voltage  due  to 

...  , .  rp,  FIG.  i.      Connections   for   magnetization 

residual    magnetism.      Ine  ,         .    , 

curve, — separately   excited. 

field  current  is  then  in- 
creased by  steps  from  zero  to  the  maximum*  rating  of  the 
machine,  the  readings  taken  at  each  step  giving  the  "  ascending  " 
curve.  The  descending  curve  is  then  obtained  by  decreasing  the 
field  current  by  steps  again  to  zero.  In  Fig.  3,  only  the  ascend- 
ing curve  is  shown ;  see  also  Fig.  2,  of  Exp.  i-B. 

To  obtain  a  smooth  curve,  the  field  current  must  be  increased 
or  decreased  continuously;  there  will  be  a  break  in  the  curve  if 
a  step  is  taken  backwards  or  if  the  field  circuit  is  broken  during 

*(§?a).  Current  Density.— For  field  windings  an  allowable  current 
density  is  800-1,000  amp.  per  sq.  in.  (1,600-1,275  circ.  mils  per  amp.)  ;  for 
armatures,  2,000-3,000  amp.  per  sq.  in.  (640-425  c.  mils  per  amp.).  For  a 
short  time  these  limits  can  be  much  exceeded.  The  sectional  area  of  a 
wire  in  circular  mils  is  the  square  of  its  diameter  in  thousandths  of  an 
inch. 


6  DIRECT   CURRENT   GENERATORS.  [Exp. 

a  run.  This  is  true  of  all  characteristics  or  other  curves  involv- 
ing the  saturation  of  iron. 

§  8.  Brush  Position. — During  the  run  the  brushes  are  kept  in 
one  position;  if  for  any  reason  they  are  changed,  the  amount 
should  be  noted.  For  a  generator  the  best  position  is  the  position 
of  least  sparking  and  of  maximum  voltage,  which  locates  the 
brushes  on  the  "  diameter  "  or  "  line  "  of  commutation.  Under 
load  this  line  is  shifted  forward  from  its  position  at  no  load,  on 
account  of  field  distortion  caused  by  armature  reaction,*  and  the 
brushes  are  accordingly  advanced  a  little  to  avoid  sparking.  As 
it  is  desirable  to  keep  the  brushes  in  the  same  position  in  taking 
all  the  curves,  with  load  or  without  load,  it  is  well  to  give  the 
brushes  at  no  load  a  little  lead,  but  not  enough  to  cause  much 
sparking. 

§  9.  Speed  Correction. — If  the  speed  varied  during  the  run, 
the  values  of  voltage  as  read  are  to  be  corrected  to  the  values  they 
would  be  at  some  assumed  constant  speed.  Since,  for  any  given 
field  current,  voltage  variesf  directly  with  speed,  this  correction 
is  simply  made  by  direct  proportion ;  each  voltmeter  reading  is 

*(§8a).  Armature  Reactions. — Armature  current  has  a  demagnetising 
effect  and  a  cross-magnetising  effect,  the  two  effects  together  being  called 
armature  reaction,  as  discussed  in  various  text  books.  The  demagnetizing 
effect  due  to  back  ampere-turns  weakens  the  field;  the  cross-magnetizing 
effect  due  to  cross  ampere-turns  distorts  the  field  (weakening  it  on  one 
side  and  strengthening  it  on  the  other)  and  shifts  forward  the  line  of 
commutation.  In  many  early  machines  this  made  it  necessary  to  shift 
the  brushes  forward  or  back  with  change  of  load  to  avoid  sparking;  in 
modern  machines  the  armature  reactions  are  not  sufficient  to  make  this 
necessary  and  the  brushes  are  kept  in  one  position  at  all  loads. 

If  a  very  accurate  determination  of  the  neutral  position  of  the  brushes 
is  desired,  it  can  be  found  by  a  voltmeter  connected  to  two  sliding  points 
which  are  the  exact  width  of  a  commutator  bar  apart.  The  neutral  posi- 
tion is  the  position  of  zero  voltage  between  adjacent  commutator  bars,  and 
this  is  shown  by  the  voltmeter. 

t  (§Qa).  If  the  speed  can  be  varied  at  will,  this  can  be  verified  for  one 
field  excitation.  A  peripheral  speed  of  3,000  feet  per  minute  is  permissible 
with  the  ordinary  drum  or  ring  armature. 


i-A] 


SERIES  GENERATOR. 


multiplied  by  the  assumed  constant  speed  and   divided  by  the 
observed  speed. 

§  10.  Curve. — After  the  speed  correction  is  applied,  the  magne- 
tization curve  is  plotted  as  in  Fig.  3.  The  abscissae  of  this  curve, 
field  amperes,  are  proportional  to  field  ampere-turns  or  magneto- 
motive force;  the  ordinates,  volts  generated  at  constant  speed, 
are  (by  Faraday's  principle,  §  I )  proportional  to  magnetic  flux. 
The  curve,  therefore,  is  a  magnetization  curve  (showing  the 
relation  between  magnetic  flux  and  magnetomotive  force)  for 
the  magnetic  circuit  of  the  generator,  which  is  an  iron  circuit 
with  an  air  gap.  The  bend  in  the  curve  indicates  the  saturation 
of  the  iron. 

§  ii.  External*    Series    Characteristic. — This   characteristic, 
which  is  the  operating  or  load  characteristic  of  the  machine,  shows 
the    variation    in    terminal 
voltage    for   different   cur- 
rents, when  the  machine  is 
self  excited  and  the  exter- 
nal   resistance    is    varied. 
The    armature,    field    and 
external     circuit     are     in 
series,  as  in  Fig.  2;  read- 
ings are  taken  of  current, 
voltage  and  speed,  for  an  ascending  curve  as  in  Fig.  3.     The 
descending  curve  may  be  taken  if  desired. 

For  any  point  on  the  curve,  the  resistance  of  the  external  cir- 
cuit is  R  —  E  -^  7,  or  the  tangent  of  the  angle  between  the  /-axis 
and  a  line  drawn  from  the  point  to  the  origin.  Below  the  knee 
of  the  curve,  it  will  be  seen  that  a  small  change  in  the  external 
resistance  will  make  a  large  change  in  current  and  voltage. 

*  (§  ua.)  In  any  characteristic  the  term  "external"  indicates  that  the 
values  of  current  and  voltage  external  to  the  machine  are  plotted;  the  term 
"  total "  indicates  that  the  total  generated  armature  current  and  voltage  are 
the  quantities  used. 


FIELD 


ARMATURE 


FIG.  2.     Connections  for  series  character- 
istic,— self  excited. 


8  DIRECT   CURRENT   GENERATORS.  [Exp. 

If  the  speed  varied  during  the  run,  the  external  characteristic 
should  be  corrected*  for  speed  as  before  (§9). 

The  watts  output,  for  any  point  on  the  external  characteristic 
is  given  by  the  product  of  current  and  voltage,  and  may  be 
plotted  as  a  curve. 

§  12.  If  the  field  coil  is  connected  so  that  the  current  from 
the  armature  flows  through  it  in  the  wrong  direction,  so  as  to 
demagnetize  instead  of  building  up  the  residual  magnetism,  the 
machine  will  not  "  pick  up."  For  one  direction  of  rotation,  the 
proper  connection  of  the  field  will  be  found  to  be  independent  of 
the  direction  of  the  residual  magnetism.  Note  the  effect  of  pre- 
vious magnetization  (from  an  outside  source)  first  in  one  and 
then  in  the  other  direction,  and  the  effect  in  each  case  of  revers- 
ing the  field  connections. 

§  13.  Total  Series  Characteristic.— The  total  characteristic  is 
derived  from  the  series  characteristic,  so  as  to  show  the  total 
generated  electromotive  force  instead  of  the  terminal  brush 
voltage. 

Resistance  Data. — The  only  additional  data  needed  are  the  volt- 
age drops  through  the  field  and  armature  for  different  currents ; 
this  is  plotted  as  a  curve  (Fig.  3)  which  is  practically  a  straightt 
line.  With  the  armature  stationary,  current  from  an  outside 
source  is  passed  through  the  field  or  armature  (separately)  ;  the 
current  is  measured  and  the  difference  of  potential  at  the  termi- 
nals. The  ratio  E  -=-  7  gives  the  resistance.  This  is  called  meas- 

*(§iib).  This  correction  is  applied  to  the  external  characteristic  and 
not  to  the  total  characteristic  for  convenience.  Inasmuch  as  it  is  the 
generated  electromotive  force  which  is  proportional  to  speed,  to  be  accurate 
the  correction  should  be  applied  to  the  total  and  not  to  the  external  char- 
acteristic. 

t  (§  T3a).  This  would  be  a  straight  line  if  the  resistance  were  constant. 
The  resistance  varies  with  temperature;  see  Appendix,  §  15.  The  armature 
resistance  also  varies  with  current  since  it  includes  the  resistance  of 
brushes  and  brush  contact,  which  depends  upon  current  density.  The  hot 
resistance  is  to  be  measured  after  the  machine  has  run  awhile,  and  is  to 
be  considered  constant. 


i-A] 


SERIES  GENERATOR. 


•^ACTERISTIC 


tiring  resistance  by  "fall  of  potential"  method;  see  Appendix, 

§17- 

Curve. — By  adding  to  the  external  characteristic  the  RI  drop 
for  field  and  armature,  we  have  the  generated  voltage  or  total 
characteristic  Fig.  3. 

Interpretation. — The  total  characteristic  falls  below  the  magne- 
tization curve  on  account  of  armature  reaction,  that  is,  the  de- 
magnetizing effect  of  the 
armature  current  which 
weakens  the  field  and 
hence  reduces  the  gen- 
erated voltage ;  for,  in 
taking  the  magnetization 
curve,  there  was  no 
armature  current  and 
hence  no  armature  reac- 
tion. The  external  char- 
acteristic falls  below  the 
total  series  characteris- 
tic, on  account  of  resist- 
ance drop. 

The  magnetization 
curve  would  be  higher 
than  the  total  character- 


FIG.    3. 


AMPERES 

Characteristics    of 
generator. 


istic  for  all  currents,  if 
in  taking  it  the  brushes 
were  given  no  lead,  that  is  were  in  the  position  of  maximum  volt- 
age. Giving  the  brushes  a  lead  lowers  the  magnetization  curve 
so  that  for  small  values  of  the  current  it  may  fall  below  the 
total  characteristic. 


10 


DIRECT   CURRENT   GENERATORS. 


[Exp. 


APPENDIX    I. 
MISCELLANEOUS  NOTES. 

§  14.  Current  and  Voltage  Adjustment. — For  currents  of  small 
values,  when  a  wide  range  of  adjustment  is  desired,  a  series  resist- 
ance (Fig.  i)  is  frequently  inadequate  and  it  is  better  to  shunt  off 
current  from  a  resistance  R,  as  in  Fig.  4. 


I 


FIG. 


FIG.  5. 


Methods  for  adjusting  voltage  or  current. 

By  adjusting  the  slider  p,  the  voltage  delivered  to  the  apparatus 
under  test  can  be  given  any  desired  value  from  zero  up  to  the  value 
of  the  supply  voltage.  A  modification  which  is  sometimes  conveni- 
ent employs  two  resistances,  B  and  C,  Fig.  5.  The  adjustment  is 
made  by  short  circuiting  or  cutting  out  more  or  less  of  one  resistance 
or  the  other,  but  not  of  both.  The  full  amount  of  one  resistance 
should  always  be  in  circuit. 

§  15.  Temperature  Corrections.— The  conductivity  of  copper  varies 
with  temperature,  according  to  the  law  given  below.  Resistance 
values  to  be  significant  should  therefore  be  for  some  specified  tem- 
perature; known  for  one  temperature  they  can  be  computed  for  any 
other.  Temperature  rise  can  be  computed  from  increase  in  resist- 
ance. In  all  cases  where  accuracy  of  numerical  results  is  important, 
as  in  commercial  tests  for  efficiency,  regulation,  etc.,  definite  tempera- 
ture conditions  should  be  obtained ;  for  this  the  detailed  recommenda- 
tions of  the  A.  I.  E.  E.  Standardization  Rules  should  be  consulted. 
To  meet  standard  requirements,  a  run  of  several  hours  is  commonly 
required.  In  practice  work  this  is  not  necessary,  it  being  usually 
sufficient  to  specify  resistances  as  cold  when  taken  at  the  beginning 
and  hot  when  taken  at  the  close  of  the  test. 

Let  Rt  be  the  resistance  of  a  copper  conductor  at  a  temperature 


i-A]  SERIES  GENERATOR.  n 

t°  C.  At  a  higher  temperature  the  resistance  will  be  greater  and 
experiment  shows  that  the  increase  in  resistance  will  be  in  direct 
proportion  to  the  temperature  rise. 

At  a  temperature  (t  -\-  6}   °C.  the  resistance  is  accordingly 


The  temperature  coefficient  a  (per  degree  C.)  depends  upon  the 
initial  temperature  t  (degrees  C),  or  the  temperature  for  which  the 
resistance  is  taken  as  100  per  cent.,  and  has  for  copper  the  following 
values  :*  — 

t         o°          6°          12°         18°         25°         32°         40°        48° 

a   .0042   .0041   .0040   .0039   .0038   .0037   .0036   .0035 

From  the  formula  given  above,  if  the  resistance  is  known  for  one 
temperature,  the  resistance  can  be  computed  for  any  other  tempera- 
ture or  for  any  temperature  rise. 

§  16.  From  this  formula  we  can  also  compute  the  temperature  rise 
6,  above  the  initial  temperature  t,  corresponding  to  a  known  increase 
in  resistance.  By  transformation  the  formula  becomes 


The  temperature  rise  above  an  initial  temperature  t  is  accordingly 
equal  to  the  per  cent,  increase  in  resistance  divided  by  a. 

§  17.  Fall  of  Potential  Method  for  Measuring  Resistance.  —  This 
method  is  based  upon  the  fact  that  the  fall  of  potential  through  a 
resistance  R  carrying  a  current  /  is  E  =  RI  (Ohm's  Law).  The 
resistance  R  v/hich  is  to  be  determined  may  be  the  resistance  of  any 
conductor  whatever  (transformer  coil,  field  winding,  armature,  etc.) 
which  will  carry  a  measurable  current  without  undue  heating  and 
is  not  itself  a  source  of  electromotive  force.  An  armature,  there- 
fore, must  be  stationary  while  its  resistance  is  being  measured  by 
this  method. 

Connect  the  unknown  resistance  to  a  source  of  direct  current 
through  a  regulating  resistance,  Fig.  6  (see  also  §  14),  so  that  the 
current  will  not  unduly  heat  the  resistance  or  exceed  the  range  of 
instruments.  Take  readings  of  the  two  instruments  simultaneously, 

*  A.  I.  E.  E.  Standardization  Rules  ;  also,  A.  E.  Kennelly,  Electrical 
World,  June  30,  1906. 


12  DIRECT   CURRENT   GENERATORS.  [Exp. 

and  without  delay  so  as  to  minimize  the  effect  of  heating.  The  re- 
sistance R  is  equal  to  E-+-L 

Fig.  6  shows  the  usual  arrangement  of  apparatus,  in  case  the  volt- 
meter current  is  but  a  small  part  of  the  total  current.  The  voltmeter 

leads  should  be  connected  di- 
rectly  to  the  resistance  to  be 
measured  (not  including  un- 
necessary connectors,  etc.) 
or  should  be  pressed  firmly 
against  its  terminals.  The 
resistance  of  an  armature 

winding  is  taken  by  pressing 
FIG.  6.     Measurement   of   resistance   by         , 

r  ,,    r  .  .       i-  •.  the   voltmeter   leads    against 

fall-of-potential  method. 

the  proper  bars  180°  or  90° 

apart;  if  resistance  of  brushes  and  connections  is  to  be  included,  the 
voltmeter  is  connected  outside  of  these  connections. 

In  case  the  ammeter  current  is  very  small,  so  that  the  voltmeter 
current  is  a  considerable  part  of  the  total  current,  the  voltmeter  should 
be  connected  outside  the  ammeter  so  as  to  measure  the  combined 
drop  of  potential  through  the  ammeter  and  unknown  resistance. 
With  the  voltmeter  connected  either  way,  an  error  is  introduced 
which  may  often  be  neglected  but  can  be  corrected  for  when  par- 
ticular accuracy  is  desired. 

§  18.  The  voltmeter  should  always  be  disconnected  before  the  cir- 
cuit is  made  or  broken,  or  any  sudden  change  is  made  in  the  current, 
to  avoid  damage  to  the  instrument. 

If  the  resistance  being  measured  is  highly  inductive,  not  only  the 
instrument  but  also  the  insulation  of  the  apparatus  under  test  may 
be  damaged  by  suddenly  breaking  the  current  through  it  on  account 
of  the  high  electromotive  force  induced  by  the  sudden  collapse  of 
the  magnetic  field.  This  may  be  avoided  by  gradually  reducing  the 
current  before  breaking  the  circuit. 

§  19.  The  value  of  an  unknown  resistance  can  be  found  in  terms 
of  a  known  resistance  placed  in  series  with  it  by  comparing  the  drops 
in  potential  around  the  two  resistances,  the  current  in  each  having 
the  same  value. 


i-B]  COMPOUND   GENERATOR.  13 

EXPERIMENT  i-B.  Characteristics  of  a  Compound*  Generator. 

§  i.  Introductory. — A  compound  generator  is  made  for  the 
purpose  of  delivering  current  at  constant  potential  either  at  the 
terminals  of  the  machine  or  at  some  distant  receiving  point  on  the 
line.  In  the  former  case  the  machine  is  flat  compounded,  the 
ideal  being  the  same  terminal  voltage  at  full  load  as  at  no  load, 
giving,  a  practically  horizontal  voltage  characteristic.  In  the 
latter  case  the  machine  is  over  compounded,  giving  a  terminal 
voltage  which  rises  from  no  load  to  full  load  to  compensate  for 
line  drop,  so  that  at  the  receiving  end  of  the  line  the  voltage 
will  be  constant  at  all  loads.  Constant  potential  service  is  used 
both  for  power  and  for  lighting.  Constant  delivered  voltage  is 
essential  in  lighting  for  steadiness  of  illumination  and  in  power 
for  constant  speed. 

§  2.  For  such  service,  the  series  generator  is  not  at  all  adapted, 
its  voltage  being  exceedingly  low  at  no  load  and,  for  a  certain 
range,  increasing  greatly  with  load. 

§  3.  A  shunt  generator  almost  meets  the  conditions,  generating 
a  voltage  which  is  nearly  constant  but  decreasing  slightly  with 
load  (Figs.  4  and  6).  Obviously  by  increasing  the  field  excita- 
tion (field  ampere-turns)  when  the  machine  is  loaded,  the  voltage 
can  be  increased  to  the  desired  value ;  this  is  true,  however,  only  in 
case  the  iron  is  not  saturated  and  it  is  accordingly  possible  for  the 
increase  in  field  ampere-turns  to  produce  a  corresponding  in- 
crease in  the  magnetic  flux.  (Compare  Fig.  2.)  In  a  shunt 
machine  this  increase  in  field  excitation  (fan  be  obtained  by  an 
increase  in  field  current  produced  either  by  an  attendant  who 
adjusts  the  field  rheostat  or  by  an  automaticf  regulator. 

*  (§ia).  This  experiment  can  be  applied  to  a  Shunt  generator  by  omit- 
ting §§20-25. 

t  (§  3a).  Tirrell  Regulator. — Many  older  forms  of  regulators,  which  oper- 
ated by  varying  field  resistance,  are  superseded  by  the  Tirrell  Regulator. 
This  regulator  operates  through  a  relay  as  follows:  (i)  When  the  volt- 
age is  too  low,  it  momentarily  short  circuits  the  field  rheostat,  causing  the 


H  DIRECT   CURRENT   GENERATORS.  [Exp. 

§  4.  In  a  compound  generator,  the  necessary  increase  of  field 
excitation  with  load  is  simply  ancf  effectively  obtained  by  means 
of  an  auxiliary  series  winding.  Since  the  current  in  the  series 
winding  is  the  load  current,  the  magnetizing  action  of  the  series 
winding  (that  is  its  ampere-turns  or  magnetomotive  force)  in- 
creases in  direct  proportion  to  the  load.  This  increases  the  mag- 
netic flux  and  hence  the  generated  voltage  by  an  amount  depend- 
ent upon  the  degree  of  saturation  of  the  iron. 

Looked  at  in  another  way,  a  shunt  winding  (which  alone  gives 
a  falling  characteristic)  and  a  series  winding  (which  alone  gives 
a  rising  characteristic)  are  combined  so  as  to  give  the  desired 
.flat  compounding  or  a  certain  degree  of  over-compounding.  As 
the  shunt  winding  -alone  gives  very  nearly  the  desired  charac- 
teristic, the  shunt  is  the  principal  winding,  the  series  winding 
being  supplementary  and  of  relatively  few  ampere-turns. 

The  characteristic  curves  for  a  shunt  or  compound  generator 
may  be  classed  as  no-load  characteristics,  and  load  characteristics. 

PART  I.    NO-LOAD  CHARACTERISTIC. 

§  5.  There  is  one  no-load  characteristic,  the  saturation  curve, 
which  shows  the  saturation  of  the  iron,  for  different  field  exci- 
tations ;  for  this  the  generator  is  usually  self-excited  but  may  be 
separately  excited  when  so  desired. 

§6.  (a)  No-load  Saturation  Curve.* — This  curve  shows  the 
terminal  voltage  for  different  values  of  field  current. 

§  7.  Data. — The  machine  is  connected  as  a  self-excited  shunt 

voltage  to  rise;  (2)  when  the  voltage  is  too  high,  it  momentarily  removes 
the  short  circuit,  causing  the  voltage  to  fall.  The  voltage  would  be  much 
too  high  or  too  low,  if  the  short  circuit  were  permanently  made  or  broken. 
The  short  circuit  is,  however,  rapidly  made  and  broken  and  of  a  varying 
duration,  a  nearly  constant  voltage  being  thus  secured.  It  may  be  applied 
directly  to  a  generator  (D.C.  or  A.C.)  or  to  its  exciter.  It  may  be  used 
advantageously  in  connection  with  a  compound  winding,  and  may  be 
arranged  so  as  to  cause  the  voltage  to  rise  with  load  in  the  same  manner 
as  in  an  over  compounded  generator. 

*Also  called  excitation  characteristic,  or  internal  shunt  characteristic. 


i-B] 


COMPOUND   GENERATOR. 


RHEOSTAT 

FIG.   i.     Connections  for  no- 
load  saturation  curve. 


generator,  Fig.  i,  and  is  driven  without  load  at  constant  speed. 
Readings  are  taken  of  field  current,  terminal  voltage  and  speed. 
The  field  current  is  varied  by  adjust- 
ing the  field  rheostat  by  steps  from  its 
position  of  maximum  to  minimum  re- 
sistance. This  gives  the  ascending- 
curve  ;  the  resistance  is  then  increased 
again  to  its  maximum  for  the  descend- 
ing curve.  If  the  rheostat,  with  re- 
sistance all  in,  does  not  sufficiently 
reduce  the  field  current,  a  second 

rheostat  may  be  placed  in  series  with  it.  The  machine  "  builds 
up  "  from  its  residual  magnetism  as  does  the  series  generator ; 
if  the  field  winding  is  connected  to  the  armature  in  the  wrong 

direction,  the  machine  will  not 
lick  up  but  will  tend  to  be- 
come demagnetized.  Should 
the  direction  of  rotation  be 
reversed,  the  field  connection 
should  be  reversed. 

§  8.  Curves. — Voltage  read- 
ings are  corrected  by  propor- 
tion for  any  variation  in  speed 
(§9,  Exp.  i-A),  and  the 
curves  plotted  as  in  Fig.  2. 

§9.  Interpretation  of 
Curves. — The  curves  in  Fig.  2 
show  the  saturation  of  the 
iron  and  are  much  the  same 
as  the  characteristic  of  a 

series  dynamo.  The  current  through  the  armature  is  small, 
being  only  a  few  per  cent,  of  full-load  current ;  the  resistance  drop 
through  the  armature  may  accordingly  be  neglected  and  the  meas- 
ured terminal  voltage  be  taken  as  (practically)  equal  to  the  total 


2  3 

FIELD  AMPERES 

FIG.  2.      No-load    saturation    curve. 


16  DIRECT   CURRENT   GENERATORS.  fExp. 

generated  voltage.  Likewise,  the  armature  current  is  so  small 
that  armature  reactions  are  negligible,  and  the  curve  is  practically 
the  same  as  a  separately-excited  magnetization  curve.  There  is 
no  necessity,  therefore,  for  taking  curves  both  self-  and  sepa- 
rately-excited. By  separately  exciting  a  generator,  it  is  pos- 
sible to  obtain  a  higher  magnetization  and  consequently  a  higher 
generated  voltage  than  can  be  obtained  by  self-excitation. 

In  design  work  and  in  manufacturing  tests,  the  saturation  curve 
is  commonly  plotted  with  field  ampere-turns,  instead  of  amperes, 
as  abscissse.  However  plotted,  the  abscissae  are  proportional  to 
magneto-motive  force  and  the  ordinates  to  magnetic  flux.* 

§  10.  Saturation  Factor  and  Percentage  of  Saturation. — There 
are  two  ways  for  expressing!  numerically  the  amount  of  satura- 
tion for  any  point  P  on  the  working  part  of  the  curve.  ( I )  The 
saturation  factor,  f,  is  the  ratio  of  a  small  percentage  increase  in 
field  excitation  to  the  corresponding  percentage  increase  in  volt- 
age thereby  produced.  (2)  The  percentage  of  saturation,  p,  is 
the  ratio  OA  -f-  OB,  when  in  Fig.  2  a  tangent  to  the  curve  at  P 
is  extended  to  A. 

Compute  these  two  for  some  one  point  on  the  curve,  corre- 

*  (§9a)-  Magnetic  Units. — For  electrical  quantities  there  are  three  sys- 
tems of  units  in  use — the  C.G.S.  electromagnetic,  the  C.G.S.  electro- 
static and  the  practical  or  volt-ohm-ampere  system.  For  magnetic  quanti- 
ties there  is  only  one  system  of  units  in  use,  the  C.G.S.  electromagnetic 
system;  magnetic  units  of  the  practical  system  would  be  of  inconvenient 
size,  they  have  no  names  and  are  never  used. 

The  unit  of  magnetic  flux  is  the  maxwell,  which  is  one  C.G.S.  line  of 
force.  The  unit  of  flux  density  is  the  gauss,  which  is  one  maxwell  per 
square  centimeter.  The  unit  of  magnetomotive  force  is  the  gilbert,  which 
is  (10-^-45)  ampere-turn.  The  unit  of  reluctance  is  the  oersted,  which 
is  a  reluctance  through  which  a  magnetomotive  force  of  one  gilbert  pro- 
duces a  flux  of  one  maxwell.  The  maxwell  and  the  gauss  are  author- 
ized by  International  Electrical  Congress,  but  not  the  gilbert  and  the 
oersted. 

Analogous  to  Ohm's  Law  (current  =  electromotive  force  -f-  resistance), 
we  have  the  corresponding  law  for  the  magnetic  circuit :  flux  (maxwells) 
=  magnetomotive  force  (gilberts)  -l- reluctance  (oersteds). 

f  A.  I.  E.  E.  Standardization  Rules,  57,  58. 


i-B]  COMPOUND   GENERATOR.  17 

spending    say    to    normal    voltage,    and    check    by   the    relation 

P  =  i  - 1//- 

These  terms  are  useful  because  they  make  possible  an  exact 
numerical  statement  of  the  degree  of  saturation  of  a  machine, 
under  working  conditions,  without  the  reproduction  of  a  satura- 
tion curve.  For  a  more  complete  study,  compute  p  and  /  for 
different  points  and  plot. 

PART  II.    LOAD  CHARACTERISTICS. 

§  II.  The  usual  load  characteristics  are  the  shunt,  compound 
and  armature  characteristics. 

In  taking  the  shunt  and  compound  characteristics,  the  machine 
is  left  to  itself  with  the  field  rheostat  in  one  position  during  the 
run,  the  curve  showing  the  variation  in  terminal  voltage  with  load. 

In  taking  the  armature  characteristic  the  field  rheostat  is  con- 
stantly adjusted;  the  curve  shows  the  variation  in  excitation 
necessary  to  maintain  a  constant  terminal  voltage  at  different 
loads. 

The  differential  and  series  characteristics  are  not  commercial 
characteristics  but  are  included  to  show  more  fully  the  operation 
of  the  series  winding.  (For  full-load  saturation  curve,  see  §  33.) 

§  12.  (b)  Shunt  Character- 
istic.— This  is  the  working 
characteristic  of  the  machine 
when  operated  at  normal 
speed  as  a  shunt-wound  gen- 
erator and  shows  the  varia- 
tion in  terminal  voltage  with 

FIG.    3.     Connections    for   shunt   cnarac- 

load  (Curve  A,  Fig.  4).  teristic> 

§  13.  Data. — The     connec- 
tions are  shown  in  Fig.  3.     Readings  are  taken  of  terminal  voltage, 
field  current,  line  current  and  speed.     No  speed  correction  is  made, 
there  being  none  which  is  simple  and  accurate.     The  field  rheostat 
is  set  in  one  position  and  no  change  is  made  in  it  during  the  run. 

3 


iS 


DIRECT  CURRENT  GENERATORS. 


[£XP: 


§  14.  The  setting  of  the  rheostat  for  commercial  testing  (§  21  a) 
is  made  for  normal  voltage  at  full  load.  For  the  purposes  of  this 
experiment,  it  is  usually  preferable  to  set  the  rheostat  for  normal 
voltage  (or  for  any  selected  value  of  voltage)  at  no  load;  in 
this  case  the  shunt,  compound  and  differential  curves,  Fig.  6,  all 
start  from  the  same  no  load  voltage.  The  load  current  is  then 
increased  from  no  load  up  to  about  25.  per  cent,  overload  and 
then  decreased,  if  so  desired,  back  to  no  load.  The  return  curve 
will  fall  a  little  below,  on  account  of  hysteresis. 

Data  are  also  to  be  taken  for  a  characteristic  starting  at  no  load 
with  a  voltage  below  normal  (§  18). 

Armature  resistance  is  measured  by  the  fall-of-potential  method, 
(§  17,  Exp.  i-A). 


0  20          40  60  80  100        120          140         160 

AMPERES 

FIG.  4.     Shunt  characteristics. 

§  15.  Curves. — The  armature  RI  drop  is  plotted  as  Curve  D  in 
Fig.  4. 

For  the  external  shunt  characteristic  (Curve  A,  Fig.,  4),  plot 
observed  line  current  as  abscissae  and  observed  terminal  voltage 
as  ordinates. 


i-B]  COMPOUND   GENERATOR.  19 

For  the  total  shunt  characteristic  (Curve  B),  plot  total  arma- 
ture current*  (line  current  plus  field  current)  as  abscissae,  and 
total  generated  voltage  (terminal  voltage  plus  armature  RI  drop) 
as  ordinates. 

§  1 6.  Interpretation  (Armature  Reactions  and  Regulation}. — 
An  ideal  characteristic  would  be  the  straight  horizontal  line,  Curve 
C,  indicating  a  constant  voltage  at  all  loads.  As  a  matter  of  fact 
the  terminal  voltage  (Curve  A)  decreases  with  load.  There  are, 
at  constant  speed,f  three  causes  for  this :  ( i )  armature  resistance 
drop,  (2)  armature  reactions  which  reduce  the  magnetic  flux 
and  (3)  decreased  field  excitation  as  the  voltage  decreases. 

The  difference  between  Curves  A  and  B  shows  the  effect  of 
(i)  resistance  drop;  the  difference  between  B  and  C  shows  the 
effect  of  (2)  armature  reaction  and  (3)  decreased  excitation, 
and  of  (4)  if  speed  varies. 

The  difference  between  B  and  C  will  show  the  effect  of  arma- 
ture reactions  (2)  alone^l  if  a  run  is  made  at  constant  excitation 
and  constant  speed,  thus  eliminating  (3)  and  (4).  This  is  the 
practical  method  for  determining  armature  reactions.  The  ma- 
chine may  be  self  excited  or  (preferably)  separately  excited. 

§  17.  The  regulation  of  the  generator  is  shown  by  the  drop 
in  Curve  A.  To  express  regulation  numerically  as  a  per  cent., 
the  rated  voltage  at  full  load  is  taken  as  100  per  cent.  In  a 
commercial  test,  therefore,  the  curve  is  taken  by  beginning  at 
full  load  at  rated  voltage  (100  per  cent.)  and  proceeding  to  open 
circuit.  The  regulation||  is  the  per  cent,  variation  from  normal 

*The  difference  between  line  and  armature  currents  is  so  small  that  for 
many  practical  purposes  the  distinction  between  them  can  be  neglected. 

t  (§i6a).  Should  the  generator  slow  down  under  load,  as  when  driven 
by  an  induction  motor,  this  would  constitute  a  fourth  cause  (4). 

$  (§  i6b).  Included,  as  a  part  of  armature  reaction,  is  the  effect  of  local 
self-induction  of  the  armature  conductors,  when  traversed  by  the  arma- 
ture current  which  (in  any  one  conductor)  is  rapidly  reversing  in  direc- 
tion. 

II  A.  I.  E.  E.  Standardization  Rules  187,  ct  seq. 


20 


DIRECT   CURRENT   GENERATORS. 


[Exp. 


full-load  voltage   (in  this  case  the  per  cent,  increase)   in  going 
from  full  load  to  no  load. 

§  1 8.  Characteristics  taken  with  Low  Field  Excitation. — On 
short  circuit  a  shunt  generator  has  no  field  excitation  and  the 
short-circuit  current  (depending  on  residual  magnetism)  is  com- 
monly less  than  normal  full-load  current.  The  current,  however, 
is  much  greater  before  short  circuit  is  reached.  On  account  of 
this  excessive  current,  the  complete  characteristic  curve  cannot 
be  obtained  with  the  field  rheostat  in  its  normal  .setting.  To 
show  the  form  of  the  complete  shunt  characteristic,  set  the  field 
rheostat  for  a  no-load  voltage  much  below  normal,  and  take 
Curve  E  (Fig.  4)  from  open  circuit  to  short  circuit,  and  Curve  F 
returning  from  short  circuit  to  open  circuit.  The  form  of  these 
curves  should  be  interpreted. 

§  19.  With  a  weak  field,  armature  reactions  cause  the  terminal 
voltage  to  fall  off  with  load  more  rapidly  than  with  a  strong 

field.  This  is  seen  by  com- 
paring Curves  E  and  F 
with  Curve  A.  The  effect 
of  armature  reactions  is 
least  when  the  iron  is 
highly  saturated,  for  then 
any  decrease  in  magneto- 
motive force  (due  to  arma- 
ture ampere-turns)  does 
not  cause  a  corresponding 
decrease  in  magnetic  flux. 

(Compare  Fig.  2.)  It  follows,  therefore,  that  a  shunt  generator 
gives  the  best  regulation  when  worked  above  the  knee  of  the 
saturation  curve.  It  will  be  found  (§  22)  that  this  is  not  so  for 
a  compound  generator. 

§  20.  (c)  Compound  Characteristic. — The  connections  for 
taking  the  compound  characteristic,  Fig.  5,  are  the  same  as  for 
the  shunt  characteristic,  Fig.  3,  with  the  addition  of  the  series 


FIG.  5.     Connections  for  compound  char- 
acteristic. 


i-B] 


COMPOUND   GENERATOR. 


21 


field  winding  which  is  in  series*  with  the  armature.  The  same 
readings  of  terminal  voltage,  field  current,  line  current  and  speed 
are  taken  as  for  the  shunt  characteristic  and  no  speedf  correc- 
tions are  made. 


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}           20           40           60           80          100         120          140         160          180         200 

AMPERES 

FIG.  6.     Series,  shunt,  compound  and  differential  characteristics. 

§  21.  In  Fig.  6  are  plotted  shunt,  compound  and  differential 
characteristics,  beginning  with  the  same  no-load  voltage.J 

The  compound  characteristic  cannot  be  made  a  perfectly 
straight  line  from  no  load  to  full  load.  What  can  be  done  is 
to  have  the  terminal,  voltage  at  full  load  the  same  as  the  no-load 

*  (§  2oa).  Short  Shunt  and  Long  Shunt. — The  connection  shown  in 
Fig.  5  is  short  shunt;  it  would  be  long  shunt  if  the  shunt  field  were  con- 
nected to  the  line  terminals  ac,  instead  of  to  the  armature  terminals  ab. 
Both  methods  of  connection  are  used  commercially,  the  difference  between 
them  being  slight. 

f  (§2ob).  A  generator  is  compounded  for  the  particular  speed  at  which 
it  is  to  operate.  When  it  is  to  be  driven  by  an  induction  motor,  it  may 
be  compounded  so  as  to  take  into  account  the  slip  of  the  motor,  i.  e., 
its  slowing  down  under  load. 

$  (§2ia).  In  commercial  testing,  the  compound  and  shunt  characteristics 
would  be  taken  with  the  same  normal  voltage  at  full  load  (§  14).  The 
differential  characteristic  would  not  be  taken. 


DIRECT   CURRENT   GENERATORS.  [Exp. 

voltage  (flat  compounding)  or  a  definite  percentage  higher  (over 
compounding).  In  either  case  the  regulation  is  the  maximum 
percentage  deviation  from  the  ideal  straight  line  at  any  part  of 
the  characteristic,  rated  full-load  voltage  being  taken  as  100 
per  cent.  (See  §  17,  and  Standardization  Rules.) 

§22.  If  the  field  magnets  of  a  compound  generator  are  highly 
saturated,  the  increase  in  field  ampere-turns  with  load  due  to  the 
series  winding  cannot  cause  a  corresponding  increase  in  the  mag- 
netic flux  and  there  will  be  considerable  deviation  from  the  ideal 
straight  line  characteristic.  A  compound  generator  accordingly 
gives  better  regulation  when  the  iron  is  below  saturation,  which 
is  opposite  to  the  conclusion  reached  for  the  shunt  generator 

(§  19). 

In  a  compound  generator  there  is  less  cause  for  sparking  and 
shifting  of  brushes  than  in  a  shunt  generator,  on  account  of 
the  strengthening  of  the  field  by  the  series  winding  under  load. 
For  fluctuating  loads,  as  railway  service,  the  compound  generator 
is  accordingly  superior  and  generally  used. 

Obviously,  on  account  of  the  series  winding,  it  is  much  worse 
to  overload  or  short  circuit  a  compound  than  a  shunt  generator. 

§  23.  Shunt  for  Adjusting  Compounding. — If  the  characteristic 
of  a  compound  generator  rises  more  than  is  desired,  there  are 
too  many  series  ampere-turns.  These  can  be  reduced  without 
changing  the  number  of  turns  by  reducing  the  current  which 
flows  through  them.  This  is  done  by  a  shunt  resistance  in  paral- 
lel with  the  series  winding.  A  generator  is  usually  given  more 
series  turns  than  are  necessary,  the  desired  amount  of  compound- 
ing being  obtained  by  adjusting  the  shunt  resistance.  This  is 
much  easier  than  changing  the  number  of  series  turns  and  makes 
it  possible  to  change  the  amount  of  compounding  at  any  time, 
even  after  the  machine  is  in  use. 

§24.  (d)  Differential  Characteristic. — The  connections  for 
this  are  the  same  as  for  the  compound  characteristic  (Fig.  5) 
except  that  the  series  field  winding  is  reversed  so  as  to  be  in 


i-B] 


COMPOUND    GENERATOR. 


opposition  to  the  shunt  winding.  The  effect  of  the  series  wind- 
ing is  now  to  decrease  (instead  of  increase)  the  magnetization 
of  the  iron,  as  the  armature  current  increases,  causing  the  volt- 
age to  fall  off  with  load  more  rapidly  than  with  the  shunt  field 
alone.  As  there  is  no  demand  for  this,  generators  with  differ- 
ential winding  are  not  used.  (In  a  motor,  Exp.  2-A,  a  differ- 
ential winding  is  useful  in  giving  constant  speed). 

§25.  (e)  Series  Characteristic. — This  characteristic  shows 
the  effect  of  the  series  winding  alone,  with  the  shunt  winding  not 
connected.  The  procedure  is  the  same  as  in  testing  a  series 
generator,  the  connections  being  as  in  Fig.  2,  Exp.  i-A. 


10000 


8000 


6000 


4000 


2000 


CONSTANT  SPEED 

CONSTANT  TERMINAL  VOLTAGE 


-7,900 


40 


80  120  160 

ARMATURE,    AMPERES 


200 


FIG.  7.  Armature  characteristic  or  field  compounding  curve,  showing  that 
at  full  load  2,200  more  ampere-turns  are  needed  than  at  no  load  for  constant 
terminal  voltage. 

§  26.  (/)  Armature  Characteristic. — This  curve  is  used  in  de- 
termining the  proper  number  of  series  turns  for  compounding  a 
generator ;  and  is  therefore  frequently  called  a  field  compounding 
curve*  It  shows,  Fig.  7,  the  variation  in  field  excitation 

*  This  has  also  been  called  an  "  excitation  characteristic,"  a  name  which 
is  ambiguous  since  it  may  be  taken  to  mean  the  saturation  curve,  §6. 


24  DIRECT  CURRENT   GENERATORS.  [Exp. 

(amperes  or  ampere-turns*)  necessary  at  different  loads  to  main- 
tain a  constant  voltage  at  the  terminals  of  a  shunt  generator 
driven  at  constant  speed.f  The  connections  are  shown  in  Fig.  3, 
readings  being  taken  of  field  current,  line  current,  terminal 
voltage  and  speed.  Separate  excitation  may  be  used  when  a 
higher  excitation  is  wanted  than  can  be  obtained  by  self-excita- 
tion. The  load  current  is  increased  from  no  load  to  about  25 
per  cent,  overload.  At  each  load,  before  readings  are  taken,  the 
voltage  is  brought  to  the  desired  constant^  value  by  adjusting 
the  field  rheostat. 

§  27.  The  rise  in  the  armature  characteristic  shows  the  in- 
crease in  ampere-turns  of  excitation  needed  to  compensate  for 
loss  in  voltage  due  to  resistance  drop,  armature  reactions,  etc. 

(§16). 

If  in  service  the  machine  is  to  be  operated  as  a  shunt  gene- 
rator, this  increase  in  excitation  can  be  obtained  by  adjusting 
the  field  rheostat  as  was  done  in  obtaining  this  curve. 

If,  however,  the  machine  is  to  be  operated  as  a  compound  gen- 
erator, this  increase  in  excitation  is  to  be  obtained  by  the  ampere- 
turns  of  the  series  winding. 

§  28.  Determination  of  Proper  Number  of  Series  Turns. — We 
know  from  the  armature  characteristic  the  additional  ampere- 
turns  of  excitation  which  must  be  provided  at  full  load  to  pro- 
duce the  desired  terminal  voltage.  We  know  also  the  amperes 
(load  current)  which  will  flow  through  these  turns  at  full  load. 
The  necessary  number  of  turns  is  accordingly  readily  found  by 
dividing  ampere-turns  by  amperes.  Thus  in  Fig.  7,  we  note  that 

*To  plot  in  ampere-turns,  the  number  of  turns  in  the  shunt  field  must 
be  known ;  see  Appendix  I.  The  number  of  turns  multiplied  by  field 
current  gives  the  number  of  field  ampere-turns. 

t  (§26a).  In  case  the  generator  is  to  be  normally  driven  by  an  induc- 
tion motor,  with  speed  decreasing  with  load,  it  should  be  so  operated  in 
taking  the  armature  characteristic.  (See  §§  i6a,  2ob.) 

$  (§26b).  The  curve  may  be  taken  for  a  voltage  which  increases  with 
load;  such  a  curve  would  show  the  series  ampere-turns  to  be  added  for 
over-compounding. 


i-B]  COMPOUND   GENERATOR.  25 

for  full  load  (200  amperes)  there  are  needed  2200  more  ampere- 
turns  excitation  than  at  no  load.  The  series  winding  will  be 
traversed  by  the  current  of  200  amperes,  and  must  accordingly 
have  ii  turns  in  order  to  make  the  required  2,200  ampere-turns. 

If  the  armature  characteristic  were  a  straight  line,  the  series 
turns  calculated  as  above  would  be  the  same  for  all  loads 
and  the  generator  could  be  compounded  so  as  to  have  perfect 
regulation  and  give  an  exactly  constant  voltage  at  all  loads.  But 
the  armature  characteristic  always  curves,  bending  more  after 
saturation  is  reached.  The  series  turns  are,  therefore,  calcu- 
lated for  one  definite  load  (full  load)  ;  for  other  loads  the  com- 
pounding will  be  only  approximately  correct  (§21). 

The  armature  characteristic  and  hence  the  proper  number  of 
series  turns  for  correct  compounding,  will  differ  for  different 
speeds  and  terminal  voltage, — an  interesting  subject  for  further 
investigation. 

APPENDIX   I. 

MISCELLANEOUS  NOTES. 

§  29.  Determining  the  Number  of  Shunt  Turns. — The  number  of 
shunt  turns  on  a  generator  can  be  more  or  less  accurately  deter- 
mined, if  the  machine  has  a  series  winding  or  a  temporary  auxiliary 
winding  with  a  known  number  of  turns. 

With  the  machine  separately  excited,  take  an  ascending  no-load 
saturation  curve,  using  the  shunt  field  winding  of  unknown  turns; 
take  a  second  similar  curve,  using  the  series  or  auxiliary  field  wind- 
ing of  known  turns.  A  comparison  of  the  two  curves  shows  that 
the  shunt  winding  requires  a  much  smaller  -current  than  does  the 
auxiliary  winding  to  give  the  same  generated  armature  voltage.  De- 
termine this  ratio  of  currents  for  equal  terminal  voltage  (found  for 
several  voltages  and  averaged)  and  suppose  it  to  be  1 :  40.  The 
shunt  turns  are  then  40  times  as  many  as  the  auxiliary  turns,  the 
ampere-turns  for  equal  terminal  voltage  being  the  same.  If  for 
example  the  auxiliary  turns  are  10,  the  shunt  turns  are  accordingly 
400. 

§  30.  The  number  of  turns  in  two  field  windings  can  be  compared 


26  '  DIRECT  CURRENT   GENERATORS.  [Exp. 

by  the  use  of  a  ballistic  galvanometer  (or  voltmeter  or  ammeter  used 
as  a  ballistic  galvanometer)  ;  the  chief  advantage  of  the  method  is 
that  it  does  not  require  facilities  for  running  the  machine.  With 
the  armature  stationary  and  the  galvanometer  connected  to  the  ter- 
minals of  one  field,  break  a  certain  armature  current  and  note  the 
throw  of  the  galvanometer.  Repeat,  breaking  the  same  armature 
current  with  the  galvanometer  connected  to  the  other  field.  The 
ratio  of  galvanometer  throws  gives  the  desired  ratio  of  field  turns. 
It  is  best  to  take  a  series  of  readings  and  average  the  results. 

§  31.  An  estimate  of  the  number  of  turns  in  a  coil  can  be  made 
from  its  measured  resistance,  size  of  wire  and  mean  length  of  turn. 
This  can  be  used  as  a  check,  but  the  method  is  commonly  only 
approximate  on  account  of  the  uncertainty  of  the  data. 

§  32.  To  Compound  a  Generator  by  Testing  with  Added  Turns. — 
The  proper  number  of  series  turns  required  to  compound  a  generator 
can  be  ascertained  by  trial  by  means  of  temporary  auxiliary  turns. 
With  the  generator  running  at  full  load,  pass  current  from  an  inde- 
pendent source  through  these  auxiliary  turns  and  adjust  this  current 
until  the  terminal  voltage  of  the  generator  has  the  desired  full-load 
voltage.  This  current  (say  220  amperes),  multiplied  by  the  number 
of  auxiliary  turns  (say  10)  through  which  it  flows,  shows  that  2,200 
extra  ampere-turns  are  needed  at  full  load.  If  the  full-load  current 
is  200  amperes,  the  generator  would  accordingly  require  n  series 
turns. 

§  33.  Full-load  Saturation  Curve. — For  obtaining  this  curve,  the 
field  excitation  is  varied  and  the  load  adjusted  at  each  reading,  so 
that  the  external  current  remains  constant  at  its  full-load  value. 
Field  currents  are  plotted  as  abscissae  and  terminal  voltages  as  ordi- 
nates.  Such  a  curve  is  to  be  taken  later  (Exp.  3~A)  on  an  alter- 
nator; it  may  accordingly  be  omitted,  in  the  present  experiment,  if 
time  is  limited. 


CHAPTER   II. 
DIRECT  CURRENT  MOTORS. 

EXPERIMENT  2-A.  Operation  and  Speed  Characteristics  of  a 
Direct  Current  Motor,  (Shunt,  Compound  and  Differential). 

PART  I.    INTRODUCTORY. 

§  i.  Generators  and  Motors  Compared. — Structurally  a  direct 
current  generator  and  a  direct  current  motor  are  alike,*  the 
essential  elements  being  the  field  and  the  armature.  The  same 
machine  may  accordingly  be  operated  either  as  a  generator  or 
as  a  motor. 

Operating  as  a  generator,  the  machine  is  supplied  with  me- 
chanical power  which  causes  the  armature  to  rotate  against  a 
counterf  or  opposing  torque;  this  rotation  of  the  armature  gen- 
erates an  electromotive  force  which  causes  current  to  flow  and 
electrical  power  to  be  delivered  to  the  receiving  circuit. 

Operating  as  a  motor,  the  machine  is  supplied  with  electrical 
power  which  causes  current  to  flow  in  the  armature  against  a 
counterf  or  opposing  electromotive  force;  this  current  creates  a 
torque  which  causes  the  armature  to  rotate  and  mechanical  power 
to  be  delivered  to  the  shaft  or  pulley. 

*(§ia).  Since  generators  are  built  in  much  larger  sizes  than  motors, 
one  generator  being  capable  of  supplying  power  for  many  motors,  there 
may  be  a  difference  in  design  due  to  size.  Moderate  size  machines,  gene- 
rators or  motors,  are  built  with  few  poles, — four  being  common  in  small 
motors.  On  the  other  hand,  very  large  machines — that  is  generators — are 
built  with  many  poles. 

In  all  direct  current  machines, — generators  or  motors — it  is  common 
practice  to  use  a  stationary  field  and  a  revolving  armature  (§  i,  Exp.  i-A). 

f  (§  ib).  There  is  no  counter  torque  in  a  generator  until  current  flows 
in  the  armature;  there  is  no  counter  electromotive  force  in  a  motor  until 
there  is  rotation  of  the  armature. 


28  DIRECT   CURRENT   MOTORS.  [Exr. 

It  is  seen  that  the  operation,  either  as  a  generator  or  as  a 
motor,  involves  (i)  the  generation  of  an  electromotive  force  and 
(2)  the  creation  of  a  torque,  both  of  which  depend  upon  funda- 
mental laws  of  electromagnetism. 

§  2.  Generation  of  Electromotive  Force. — An  electromotive 
force  is  generated  in  a  generator  or  in  a  motor  due  to  the  cutting 
of  lines  of  force,  this  electromotive  force  being  proportional  to 
the  rate  at  which  the  lines  of  force  or  flux  are  cut,  as  already 
discussed  in  §  I,  Exp.  i-A. 

In  a  generator  this  electromotive  force  causes  (or  tends  to 
cause)  a  current  to  flow;  in  a  motor,  it  is  a  counter  electromotive 
force  and  opposes  the  flow  of  current. 

§  3.  Creation  of  Torque. — A  torque  is  created  in  a  generator 
or  motor  due  to  the  forces  acting  upon  a  conductor  carrying 
current  in  a  magnetic  field.  In  a  motor  this  torque  causes  (or 
tends  to  cause)  a  rotation  of  the  armature  with  respect  to  the 
field ;  in  a  generator,  it  is  a  counter  torque  and  opposes  the  rota- 
tion of  the  armature. 

The  creation  of  torque  depends  upon  the  following  funda- 
mental principle : — When  a  conductor  carrying  current  is  located 
in  a  magnetic  field,  it  is  acted  upon  by  a  .force  that  tends  to  move 
the  conductor  in  a  direction  at  right  angles  to  itself  and  to  the 
magnetic  flux,  the  force  being  proportional*  to  the  current  and 
to  the  flux  density. 

This  force  creates  a  torque, — that  is  a  turning  moment  or 
couple — equalf  to  the  product  of  the  force  and  the  length  of  the 

*  (§3a).  In  C.G.S.  units  this  force  is  equal  to  the  product  of  the  cur- 
rent, flux  density,  length  of  conductor  and  sine  of  the  angle  between  the 
conductor  and  direction  of  flux.  This  sine  is  unity  when  the  conductor 
and  flux  are  at  right  angles,  as  in  most  electrical  machinery.  When  there 
are  a  number  of  conductors,  each  conductor  is  subject -to  this  force;  the 
total  torque  of  a  motor  is  therefore  proportional  to  the  total  number  of 
armature  conductors. 

t  (§3b).  Torque  may  be  expressed  as  pounds  at  one  foot  radius,  pound- 
feet,  kilogram-meters,  etc.  Power  is  proportional  to  the  product  of  torque 


2-A]  SPEED   CHARACTERISTICS.  29 

radius  or  lever  arm  to  which  the  force  is  applied.  It  accordingly 
follows  that:  torque  is  proportional  to  armature  current  and  to 
the  flux  density  of  the  field;  this  is  irrespective  of  whether  the 
armature  is  rotating*  or  not.  A  reversal  of  either  the  current 
or  the  flux  alone  reverses  the  direction  of  the  torque. 

Of  the  total  torque,  part  is  used  in  overcoming  friction,  wind- 
age and  core  loss ;  the  remainder  is  useful  torque  and  is  available 
at  the  pulley. 

§  4.  Automatic  Increase  of  Current  with  Load. — The  counter- 
electromotive  force  E'  is  always  a  few  per  cent,  less  than  the  sup- 
ply voltage  E.  The  difference  is  due  to  the  resistance  drop  in 
the  motor  armature, — including  brushes,  brush  contact  and  con- 
nections, and  the  series  field  (if  any)  ;  that  is 

E'  =  E-RI.  (i) 

and  speed;  thus,  if  R.P.M.  is  revolutions  per  minute  and  T  is  torque  in 
pound-feet 


33,000 

If  power  is  known,  torque  may  be  found  by  dividing  power  by  speed. 
In  pound-feet,  torque  is 

33,000        H.P. 

27T         A  R.P.M.   ' 

When  power  is  in  watts,  it  is  frequently  convenient  to  express  torque  in 
synchronous  watts  ";  thus, 


"R.P.M.* 

(One  synchronous  watt  =  7.04  pound-feet.) 
(One  pound-foot  =  0.142  synchronous  watt.) 
Torque  is  also  expressed  in  "watts  at  1,000  R.P.M.";  thus, 


*(§3c).  Torque  with  the  armature  at  rest  (static  torque)  can  be  de- 
termined for  various  field  currents  and  for  various  armature  currents  by 
means  of  a  lever  arm  attached  to  the  armature  and  a  spring  balance  or 
platform  scales. 


3°  DIRECT   CURRENT   MOTORS.  [Exp. 

Within  the  usual  range  of  operation,  this  RI  drop  for  a  com- 
mercial motor  is  only  a  few  per  cent,  of  the  total  line  voltage. 
Good  design  does  not  permit  more,  inasmuch  as  the  output  and 
efficiency  are  decreased  by  the  same  percentage. 

The  current  which  flows  in  the  armature  is  seen  to  be 


If  under  running  conditions  the  current  /  is  not  sufficient  to 
give  the  motor  enough  torque  (which  is  proportional  to  current 
and  flux)  to  do  its  work  at  the  speed  at  which  it  is  running,  the 
motor  will  begin  to  slow  down,  thus  decreasing  the  counter- 
electromotive  force  E'  (which  is  proportional  to  speed  and  flux). 
As  Ef  decreases  /  increases,  until  the  torque  is  sufficient  to  meet 
the  demands  upon  the  motor.  The  current  accordingly  increases 
automatically  with  the  load,  and  this  increase  can  be  continued 
until  the  safe*  limit,  determined  by  heating,  is  reached. 

On  the  other  hand,  if  the  current  I  is  more  than  is  needed  to 
give  the  torque  required  for  the  load  at  a  certain  running  speed, 
the  surplus  torque  will  cause  the  armature  to  accelerate,  thus 
increasing  E'  and  decreasing  /  to  a  value  which  gives  the  proper 
torque  for  the  load  and  speed. 

It  will  be  seen  that  a  small  change  in  E'  is  sufficient  to  cause  a  large 
change  in  /  and  therefore  in  the  torque.  As  an  example,  suppose  E'  =  100, 
E=  104;  if  an  increase  in  speed  causes  E'  to  increase  2  per  cent.,  that  is 
to  102,  the  current  /  will  be  reduced  50  per  cent. 

§  5.  Relations  between  Speed,  Flux  and  Counter-electromo- 
tive Force.  —  Counter-electromotive  force  is  proportional  to  speed 
(S)  and  flux  (<j>)  ;  that  is 

E'cc+S.  (3) 

Hence,  speed  varies  directly  as  the  counter-electromotive  force 
and  inversely  with  the  flux  ;  that  is 

*  A  motor  is  usually  rated  so  that  it  can  be  run  for  several  hours  at 
25  per  cent,  over  its  rated  load. 


2-A]  SPEED   CHARACTERISTICS.  31 

E' 

5oc^r;  (4) 

id      or, 

E-  Rf 
•*«—£—•  (5) 

This  is  the  speed  equation  for  a  motor.  It  is  seen  that  if  <f> 
is  reduced  the  speed  will  increase.  The  equation  shows  the  defi- 
nite numerical  relations  of  the  quantities  involved.  Hozv  an 
increase  in  speed  is  brought  about  by  a  decrease  in  flux  is  made 
more  clear  in  §  7. 

§  6.  Speed  of  a  Shunt  Motor. — A  shunt  motor  with  constant 
supply  voltage  has  a  constant  field  current  and  therefore  a  con- 
stant flux.  It  accordingly  follows  that  the  speed  is  nearly  con- 
stant. The  RI  drop  causes  it  to  decrease  with  load  (compare 
equation  5)  ;  this  is  partially  offset,  however,  by  the  effect  of 
armature  reactions,  as  seen  later  (§8). 

§7.  It  is  seen  from  equation  (5)  that  the  speed  may  be  in- 
creased or  decreased  by  weakening  or  strengthening  the  field. 
The  process  is  explained  as  follows: — 

When  the  field  is  weakened  the  counter-electromotive  force  is 
reduced ;  this  permits  more  current  to  flow  in  the  armature,  thus 
giving  greater  torque*  and  speed.  The  speed  accordingly  in- 
creases until  E'  has  increased  so  as  to  limit  the  current  (and 
hence  the  torque)  to  a  value  which  will  give  no  further  accel- 
eration. 

The  cause  for  increase  of  speed  is  surplus  torque. 

§  8.  Armature  Reactions  and  Brush  Position. — If  the  brushes 
are  given  a  backward  lead  (which  is  usual  in  motors  running  in 
one  direction,  in  order  to  obtain  better  commutation)  the  field  is 

*  (§7^).  As  an  example,  suppose  the  field  is  weakened  so  that  the  flux 
is  reduced  2  per  cent,  and  E'  the  same  amount ;  and  suppose  the  armature 
current  increases  50  per  cent.  Torque  is  proportional  to  flux  and  armature 
current  and  in  this  assumed  case  is  increased  47  per  cent. ;  for  .98  x  1.50 
-  1-47. 


32  DIRECT   CURRENT   MOTORS.  [Exp. 

weakened  by  the  demagnetizing  effect  of  armature  reactions 
(§8a,  Exp.  i-A).  This  causes  the  flux  to  decrease  with  load, 
so  that  the  speed  does  not  decrease  as  much  as  it  would  with  the 
brushes  in  the  neutral  position.  On  account  of  armature  reac- 
tions, therefore,  the  speed  regulation  of  a  motor  is  better;  the 
voltage  regulation  of  a  generator  is  worse  (§  16,  Exp.  I— B). 

The  proper  brush  position  for  best  commutation  is  the  posi- 
tion which  gives  minimum  speed. 

§  9.  If  the  backward  lead  of  the  brushes  is  increased,  the  speed 
of  the  motor  under  load  can  be  increased  until  it  equals  or  ex- 
ceeds the  speed  at  no  load.  Such  a  control  of  speed  by  brush 
adjustment  is  not  practicable,  however,  on  account  of  bad  com- 
mutation and  destructive  sparking;  the  brushes  should  be  given 
the  position  of  best  commutation.  A  small  variation  of  speed  can 
be  made,  if  desired,  by  shifting  the  brushes,  provided  it  is  not 
enough  to  cause  much  sparking. 

§  10.  Speed  Control. — From  equation  (5)  it  is  seen  that  the 
speed  of  a  motor  can  be  varied :  by  changing  the  impressed  volt- 
age, E;  by  varying  resistance,  R  (series  controller)  ;  or,  by  vary- 
ing flux,  (j>.  Each  of  these  methods  is  in  use  for  operating 
variable  speed  motors. 

(a)  Varying  line  voltage.     Several  line  voltages  can  be  ob- 
tained by  using  a  number  of  line  wires.     Such  a  system  is  called 
a  multiple-voltage  system. 

(b)  Varying  resistance.     The  series  controller  is  in  common 
use  with  series  motors ;  it  is  used  occasionally  with  shunt  motors 
of  small  size. 

(c)  Varying   flux.     This    can   be    accomplished    either   by   a 
change   in   excitation    (magnetomotive    force)    or   a   change   in 
reluctance;  for  flux  —  magnetomotive  force  -f-  reluctance. 

( i )  Speed  control  by  varying  excitation  is  obtained  in  a  shunt 
motor  by  a  rheostat  in  series  with  the  field  (§7)  ;  in  a  series 
motor,  by  an  adjustable  resistance  in  parallel  with  the  field. 


2-A] 


SPEED    CHARACTERISTICS. 


(The  possible  method  of  control  by  brush-shifting,  §  8,  is  not 
used.) 

(2)  Speed  control  by  varying  reluctance  is  obtained  in  certain 
shunt  motors  by  varying  the  air-gap. 

A  limit  to  speed  control  by  a  variation  in  flux  (by  varying 
either  excitation  or  reluctance)  is  reached  on  account  of  arma- 
ture reactions ;  a  considerable  reduction  in  flux  causes  bad  com- 
mutation. For  varying  the  speed  through  a  wide  range,  there- 
fore, these  methods  can  only  be  used  if  the  effects  of  armature 
reactions  are  overcome. 

This      Was       first       SatisfaC-  Constant  Potential  Supply 

torily  accomplished  by  the 
compensated  winding  of 
Prof.  H.  J.  Ryan,  which 
was  placed  in  slots  in  the 
pole  faces.  This  compensa- 
tion is  now  generally  ac- 
complished by  the  more 
easily  constructed  inter- 
poles  or  commutating  'poles 
of  the  inter  pole  motor. 


o 


RHEOSTAT 
/^~^(  SHUNT  FIELD  S       N. 

— /    «rj     r-/inffitftfWtf>> /   A  y — 


STARTING  BOX 


FIG.  i.     Connections      for     operating 
shunt    motor. 


PART  II.    OPERATION. 
§11.  Shunt    Motor.— If 

the  motor  is  compound,  cut 
the   series  coil  out  of  the 
circuit.     Connect   the   sup- 
ply lines  to  the  main  terminals  of  the  motor  and  complete  the 
connections,  as  in  Fig.   I.     Note  the  queries,  §   15. 

To  start  the  motor,  have  all  the  starting  box  resistance  in  cir- 
cuit and  all  the  field  rheostat  out  of  circuit;  make  sure  that  the 
field   circuit   is   complete.     The   circuits   should  be   so   arranged 
that  closing  the  supply  circuit  will  excite  the  field   (which  takes 
4 


34  DIRECT   CURRENT   MOTORS.  [Exp. 

an  appreciable  time)  before*  the  armature  circuit  is  closed.  The 
armature  circuit  is  then  closed  and  this  is  commonly  done  by  the 
starting  box  lever. 

Bring  the  motor,  unloaded,  up  to  speed  by  cutting  the  starting 
box  resistance  slowly^  out  of  circuit  until  the  whole  resistance 
is  cut  out.  Note  the  ammeter  during  the  process  and  the  in- 
crease of  speed  as  indicated  by  the  hum  of  the  motor.  The 
starting  box  should  be  kept  in  circuit  only  during  starting,  for 
(except  in  special  cases)  it  is  not  designed  for  continuous 
operation. 

If  the  motor  does  not  now  run  at  normal  speed,  the  speed  can 
be  increased  by  gradually  varying  the  field  current  by  means  of 
the  field  rheostat.  Do  not  reduce  the  field  current  too  much, 
nor  under  any  circumstances  break  the  field  circuit,^  or  the  motor 
will  run  at  a  dangerous  speed. 

Note  the  speed  at  no  load  for  several  excitations ;  also,  when 
facilities  permit,  for  several  supply  voltages.  (For  example, 
operate  a  no-volt  motor  with  55  volts  on  armature  and  on  field; 
with  55  volts  on  armature  and  no  volts  on  field;  but  not  with 
no  volts  on  armature  and  55  volts  on  field.) 

§  12.  Stopping. — Motors  are  commonly  stopped  by  opening 
the  supply  switch  and  not  by  first  opening  the  armature  circuit. 

*  (§  na).  If  the  starting  box  were  made  with  sufficiently  high  resistance, 
so  as  to  properly  limit  the  current  irrespective  of  counter-electromotive 
force,  the  armature  circuit  could  be  closed  simultaneously  with  the  field. 
This,  however,  is  not  usual  practice. 

t  (§nb).  Starting  boxes  are  sometimes  made  so  that  it  is  impossible  to 
manipulate  them  too  rapidly.  The  "  multiple-switch  "  motor  starter,  used 
particularly  in  starting  large  motors,  has  a  number  of  switches,  thrown 
successively  by  hand;  these  give  good  contact  for  large  currents  and  re- 
quire time  for  cutting  out  the  successive  sections  of  the  resistance. 

$  (§  nc).  Automatic  Release. — This  danger  is  commonly  guarded  against 
by  a  solenoid  on  the  starting  box  which  releases  the  lever  and  allows  it  to 
spring  back  to  the  starting  position  when  there  is  no  current  in  the  field 
circuit.  This  also  acts  as  a  "  no-voltage  "  release,  giving  protection  against 
damage  which  might  occur  were  the  current  supply  cut  off  and  put  on 
again  with  the  starting  box  resistance  all  out. 


2-A]  SPEED    CHARACTERISTICS.  35 

There  is  then  no  sudden  discharge  of  field  magnetism  and  con- 
sequent liability  to  damage ;  for,  as  the  armature  slows  down  it 
generates  a  gradually  decreasing  electromotive  force  which  main- 
tains the  field  excitation  so  that  it  too  decreases  gradually.  (If 
there  is  an  automatic  release  on  the  starting  box,  it  opens  the 
armature  and  field  circuits  after  the  field  excitation  has  decreased 
to  a  low  value.) 

The  effects  of  induced  electromotive  force  caused  by  sudden 
field  discharge  can  be  reduced  by  absorbing  its  energy  in  a  high 
resistance  shunt  in  parallel  with  the  field  circuit,  or  in  a  short- 
circuited  secondary  circuit  around  the  field  core.  A  brass  field- 
spool  will  act  in  this  way. 

Throwing  power  suddenly  off  the  line,  by  opening  the  supply 
switch,  may  cause  fluctuations  in  line  voltage, — particularly  in 
case  of  large  motors  under  load.  To  avoid  this,  before  the  sup- 
ply switch  is  opened,  the  starting  resistance  may  first  be  gradu- 
ally introduced  into  the  armature  circuit,  which,  however,  is  not 
to  be  opened ;  then  the  supply  switch  is  opened. 

§  13.  Compound  Motor. — In  a  compound*  motor,  the  series 
winding  strengthens  the  field  as  the  armature  current  increases. 
On  starting  or  under  heavy  load  (i.  e.,  at  times  when  the  arma- 
ture current  is  large)  the  motor  is  accordingly  given  a  very 
strong  field  and  therefore  has — for  a  given  armature  current — 
a  greater  torque  than  it  would  have  with  the  shuntf  winding 
only. 

*  (§  I3a).  To  tell  whether  a  series  winding  is  connected  "compound  "  or 
"differentially,"  throw  off  the  belt  and  start  the  motor  (for  a  moment) 
with  the  series  coil  only.  If  the  motor  tends  to  start  in  the  same  direction 
as  it  does  with  the  shunt  coil,  the  winding  is  "  compound  "  or  "cumula- 
tive;" if  in  the  reverse  direction,  the  winding  is  "differential." 

t(§i3b).  This  means  a  greater  torque  than  it  would  have  with  the 
same  shunt  winding  only.  The  motor  could  be  given  a  different  shunt 
winding  which  would  give  as  strong  a  field  and  as  great  a  torque  as  is 
obtained  by  means  of  the  compound  winding.  Such  a  shunt  winding, 
however,  would  give  the  strong  field  at  all  times;  whereas  the  compound 
winding  gives  the  strong  field  only  at  particular  times, — i.  e.,  at  starting 
and  under  load. 


36  DIRECT   CURRENT   MOTORS.  [Exp. 

Under  load  the  compound  winding,  by  strengthening  the  field, 
causes  the  motor  to  slow  down.  For  certain  kinds  of  service — 
as  in  operating  rolling  mills,  cranes,  elevators,  etc. — this  is  desir- 
able in  that  the  motor  can  work  at  great  overload  without  the 
excessive  demand  for  power  which  would  be  made  by  a  constant 
speed  motor.  As  compared  with  a  shunt  motor,  it  works  under 
load  at  greater  torque  and  less  speed,  and  can  stand  a  greater 
overload.  In  this  respect  it  is  similar  to  the  series  motor  (see 
§  18).  It  differs  from  the  series  motor  in  that  at  light  load 
there  is  still  a  certain  strength  of  field  due  to  the  shunt  winding, 
and  the  speed,  therefore,  cannot  exceed  a  certain  value,  whereas 
a  series  motor  will  attain  a  dangerous  speed  if  the  load  is  thrown 
off.  Under  some  operating  conditions  the  compound  motor  can 
accordingly  be  used  where  neither  the  shunt  nor  the  series  motor 
would  be  suitable. 

If  slowing  down  with  load  is  not  wanted  and  a  constant  speed 
is  desired  at  all  loads,  together  with  a  large  torque  at  starting, 
the  series  winding  is  used  during  starting  only  and  is  then  cut 
out  or  short-circuited. 

§  14.  Differential  Motor. — Since  a  differential  winding  weak- 
ens the  field  as  the  load  increases,  such  a  winding  makes  possible 
a  speed  which  increases  with  load.  This  is  practically  not  desir- 
able. In  some  cases,  however,  it  is  desirable  to  have  the  same 
speed  at  full  load  as  at  no  load  and  to  use  a  series  winding  just 
sufficient  to  overcome  the  tendency  which  a  shunt  motor  has  to 
slow  down  with  load.  If  the  series  turns  are  too  many  for  this, 
their  effect  can  be  cut  down  by  a  shunt  of  proper  resistance  con- 
nected in  parallel  with  the  series  winding. 

The  starting  torque  of  a  differential  motor  is  poor,  particularly 
under  load,  inasmuch  as  the  large  starting  current  in  the  differ- 
ential winding  greatly  weakens  the  field.  For  this  reason,  when 
a  differential  winding  is  used,  it  is  usually  cut  out  of  circuit  or 
short-circuited  during  starting. 


2-A]  SPEED   CHARACTERISTICS.  37 

If  there  are  many  series  turns  and  no  shunt  is  used,  the  cur- 
rent taken  by  a  differential  motor  may  become  excessive  as  the 
load  increases,  thus  weakening  the  field  so  that  the  motor  races, 
or  even  reversing  the  field  so  that  the  motor  suddenly  reverses. 

§  15.  Queries. — For  increasing  the  speed,  is  the  field  current  increased 
or  decreased?  Why?  What  is  the  use  of  the  starting  box?  In  starting, 
why  do  you  not  close  the  field  and  armature  circuits  simultaneously?  Why 
is  the  starting  box  connected  in  series  with  the  armature  and  not  in  series 
with  the  line?  Why  is  a  strong  field  needed  for  starting?  Does  this 
become  of  more  or  of  less  importance  when  starting  under  load?  Would 
an  added  series  winding  be  an  advantage  or  a  disadvantage  in  starting? 
Why  would  it  be  dangerous  to  break  the  field  circuit?  What  is  the  effect 
of  shifting  the  brushes?  What  is  the  proper  position  for  the  brushes? 
What  is  effect  of  interchanging  positive  and  negative  supply  lines?  What 
changes  in  connections  are  necessary  to  reverse  the  direction  of  rotation 
of  the  armature?  (Be  careful  not  to  run  more  than  a  moment  in  the 
reverse  direction,  if  the  brushes  would  thus  be  damaged.) 

PART  III.    SPEED  CHARACTERISTICS. 

§  1 6.  Shunt,  Compound  and  Differential  Motor. — It  is  the  pur- 
pose of  the  experiment  to  determine  the  variation  of  speed  with 
load  for  the  same  motor  connected  in  three  ways, — shunt,  com- 
pound and  differential ;  the  line  voltage  is  constant  throughout 
the  three  runs.  The  brushes  should  be  in  one  position  during 
all  the  runs  (§8),  or  the  amount  of  any  change  noted. 

With  the  motor  connected  as  a  shunt  machine,  Fig.  I,  adjust 
the  field  current  by  means  of  the  field  rheostat  so  that  the  motor 
runs,  on  no  load,  at  the  speed  for  which  it  is  designed,  and  keep 
the  field  current  constant  at  this  value  during  the  run.  For  the 
other  two  runs,  compound  and  differential,  adjust  the  field  cur- 
rent for  this  same  no-load  speed*  and  keep  the  field  current 
constant  during  each  run. 

*  (§  i6a).  Starting  with  the  same  no-load  speed,  and  making  runs  from 
no  load  to  full  load,  gives  the  three  speed  characteristics  of  Fig.  2  coincid- 
ing at  no  load ;  this  is  the  best  procedure  for  instruction  purposes. 

In  commercial  testing,  the  field  should  be  adjusted  so  that  the  motor 
runs  at  rated  speed  at  full  load.  The  curve  is  then  taken  from  full  load 
to  no  load ;  the  maximum  per  cent,  variation  in  speed  from  its  full  load 
value  is  the  per  cent,  speed  regulation.  (Standardization  Rules,  195.) 


DIRECT    CURRENT    MOTORS. 


[Exp. 


Vary*  the  load  on  the  motor  by  steps  between  no  load  and  25 
per  cent,  overload,  reading  line  voltage,  field  current,  armaturef 
current  (or  else  line  current)  and  speed,  for  each  step.  Make 
runs  with  the  motor  connected  shunt,  compound  and  differential. 

With  current  as  abscissae  (either  line  current  or  armature 
current)  and  speed  as  ordinates,  plot  speed  characteristics  for 
the  three  runs  as  in  Fig.  2. 


1100 

1000 

900 

800 

700 

Q600 

UJ 

iu500 

ft 

400 
300 
200 
100 


mcFERENTjAL, 


04  8  12          16          20          24          28          32          36          40 

ARMATURE  CURRENT 
FIG.  2.     Speed  characteristics  of  a  motor, — shunt,  compound  and  differential. 

§  17.  It  is  instructive  to  take  runs  as  a  differential  motor  with 
different  resistances  in  shunt  with  the  series  coil ;  also,  to  take 
the  various  runs  (shunt,  compound  and  differential)  with  the 
field  excitation  above  and  below  saturation. 

*  (§i6b).  This  may  be  done  by  means  of  a  brake,  a  blower,  a  belted 
generator  or  other  convenient  load ;  if  a  generator  is  used,  its  output  may 
be  absorbed  in  resistance  or  pumped  back  into  the  line  (§26,  Exp.  2-B). 

f  (§i6c).  If  the  armature  current  is  measured,  the  field  current  is 
added  to  give  the  line  current;  if  the  line  current  is  measured,  the  field 
current  is  subtracted  to  give  the  armature  current. 


2-A] 


SPEED   CHARACTERISTICS. 


39 


APPENDIX   I. 

SERIES  MOTOR. 

§  18.  Operation. — A  series*  motor  is  distinctly  a  variable  speed 
motor.  Its  characteristics  are  shown  in  Fig.  3.  The  speed  increases 
rapidly  as  the  load  is  decreased,  becoming  dangerouslyf  great  if  the 
load  is  removed  or  reduced  too  much.  The  series  motor,  therefore, 
cannot  be  run  at  no  load 
and  normal  voltage;  it 
can  be  run  at  no  load 
with  a  series  resistance 
in  circuit. 

The  series  motor,  be- 
sides being  used  for  trac- 
"  tion,:}:  is  used  for  hoists, 
etc.  For  such  service  it 
is  well  adapted.  The  im- 
portant characteristic  is 
that  by  slowing  down 
under  heavy  load,  it  can 
increase  its  torque  with- 
out requiring  a  corresponding  increase  in  power ;  for  torque  =  power 
-r- speed  (§  3b).  If  the  speed  did  not  decrease  with  load,  it  is  seen 
that  the  power  would  have  to  be  greatly  increased  to  give  the  same 
torque.  This  would  require  a  much  larger  motor. 

*  (§  i8a).  For  the  purpose  of  comparison  with  the  shunt,  compound  and 
differential  motor,  the  characteristics  of  the  series  motor  are  here  described, 
although  its  test  is  not  usually  to  be  included  as  a  part  of  the  present 
experiment.  When  the  test  is  made,  it  is  well  to  combine  it  with  efficiency 
measurements,  §  33,  Exp.  2-B. 

t(§i8b).  In  the  laboratory,  be  prepared  to  shut  down  quickly  if  ex- 
cessive speed  is  reached.  With  a  belted  load,  there  is  danger  of  the  belt 
flying  off;  with  a  brake,  there  is  danger  of  an  unintentional  sudden  de- 
crease in  load. 

$(§i8c).  In  traction,  the  controller  is  usually  so  arranged  that  two 
motors  can  be  connected  in  series  or  in  parallel  with  each  other  for  speed 
control,  thus  giving  each  motor  half  or  full  voltage.  The  series  resistance 
is  likewise  used  for  control  and  for  starting.  In  starting,  the  resistance 
and  both  motors  are  all  in  series. 


AMPERES 

FIG.  3.     Characteristics    of    a    series 
operated    at    constant    voltage. 


motor, 


4°  DIRECT   CURRENT   MOTORS.  [Exp. 

§19.  Torque. — Since  torque  varies  as  flux  X  current,  the  torque 
would  vary  as  /2,  if  flux  were  proportional  to  current.  For  small 
currents — below  saturation — this  is  more  or  less  true.  For  large 
currents — after  saturation — the  flux  is  practically  constant  and  the 
torque  directly  proportional  to  /.  The  torque  curve,  Fig.  3,  is  there- 
fore at  first  more  or  less  parabolic  and  then  becomes  a  straight  line. 

§20.  Speed. — From  equation  (5)  it  is  seen  that  speed  varies  in- 
versely with  flux.  For  small  currents,  if  we  consider  RI  negligible 
and  flux  proportional  to  current,  speed  varies  as  1/7;  the  speed  curve 
(Fig.  3)  would  then  be  an  hyperbola.  For  larger  currents  satura- 
tion is  reached,  the  flux  becomes  practically  constant  and  the  speed 
more  nearly  constant.  On  account  of  RI  drop,  speed  continues  to 
gradually  decrease  as  current  increases,  even  after  saturation  is 
reached.  Series  motors  are  sometimes  overwound,  that  is,  wound 
so  that  saturation  (and  hence  more  constant  speed)  is  soon  reached. 

§21.  Test. — The  load  is  varied  between  an  overload  (determined 
by  maximum  safe  current)  and  an  underload  (determined  by  maxi- 
mum safe  speed).  The  line  voltage  is  constant;  a  series  resistance 
is  used  for  starting  and  may  be  used  for  adjusting  voltage.  Any 
method  for  loading  can  be  used.  If  a  shunt  generator  is  used  as  a 
load,  its  output  may  be  absorbed  in  resistance  or  pumped  back  into 
the  line.  (See  §26,  Exp.  2-B.)  The  pumping  back  method  has 
been  modified  by  A.  S.  McAllister,  so  as  to  form  a  convenient  method 
for  determining  the  torque  of  any  kind  of  motor,  direct  or  alternating 
(Standard  Handbook,  3-239  and  8^-151;  McAllister's  Alternating 
Current  Motors,  p.  185). 


2-B]  EFFICIENCY.  41 

EXPERIMENT  2-B.  Efficiency  of  a  Direct  Current  Motor*  (or 
Generator)  by  the  Measurement  of  Losses. 

§  I.  Introductory. — Efficiency  is  the  ratio  of  output  to  input. 
The  obvious  and  direct  method  for  determining  the  efficiency  of 
a  motor  is,  therefore,  to  measure  the  outputf  and  the  input  and 
take  their  ratio.  An  indirect  method,  known  as  the  method  of 
losses  or  stray  power  method,  avoids  the  measurement  of  output. 
In  this  method  the  losses  are  measured  and  the  output  obtained 
by  subtracting  the  losses  from  the  input ;  the  efficiency  is  then 
determined. 

This  method  of  losses  possesses  several  advantages  over  meth- 
ods that  involve  the  measurement  of  output.  The  motor  output 
is  in  some  cases  a  troublesome  quantity  to  measure,  especially  if 
accuracy  is  essential ;  but,  even  with  the  same  degree  of  accuracy 
in  the  measurement  of  output  and  of  losses,  the  efficiency  cannot 
be  as  accurately  determined^  from  the  former  as  from  the  latter. 

*With  the  appendices,  this  experiment  covers  the  main  features  of  the 
usual  methods  for  determining  the  efficiency  of  any  machine,  direct  or 
alternating.  The  main  experiment  is  explicit  for  determining  the  efficiency 
of  a  shunt  motor,  and  it  is  suggested  that  the  student,  without  reference 
to  the  Appendices,  first  performs  this  main  experiment.  The  Appendices 
should  then  be  read  and,  if  desired,  a  second  experiment  made  (either 
now  or  later)  under  some  of  the  special  conditions  which  are  there  treated. 

t(§ia).  Direct  Measurement  of  Output. — The  output  of  a  motor  can 
be  determined  directly  by  electrical  measurement  (using  for  a  load  a 
calibrated  generator,  §24),  or  by  mechanical  measurement  (measuring 
torque  by  means  of  a  Prony  brake,  Brackett  cradle  dynamometer,  etc.). 
Power  can  be  readily  computed  when  torque  and  speed  are  known  (§3b, 
Exp.  2-A).  There  are  various  forms  of  absorption  and  transmission 
dynamometers  conveniently  arranged  for  the  direct  measurement  of  power. 
For  description  of  Prony  brake,  see  Flather's  Dynamometers  and  the 
Measurement  of  Power  and  the  usual  hand  and  text  books ;  also  Electric 
Journal,  I.,  419.  For  the  cradle  dynamometer,  see  Nichols'  Laboratory 
Manual,  Vol.  II.,  and  elsewhere. 

$(§ib).  Let  us  suppose  that  the  error  in  measuring  the  input,  output 
or  losses  is  one  per  cent.,  due  to  inaccuracies  in  the  instruments  or  in 


42  DIRECT    CURRENT    MOTORS.  [Exp. 

t 
A  further  advantage  of  this  method  is  that  a  load  run  is  not 

an  essential,  as  will  be  seen  later,  and  hence  may  be  omitted. 
Conditions  often  arise,  as  in  testing  large  machines,  when  a  load 
test  is  impossible  and  this  advantage  then  becomes  important. 
It  is  always  best,  however,. to  make  the  load  run  when  this  can 
be  done. 

The  method  of  losses  is  general  and  can  be  applied  for  deter- 
mining the  losses,  and  hence  the  efficiency,  of  a  shunt,  compound, 
differential  or  series  wound  motor  or  generator.  In  the  follow- 
ing paragraphs  the  directions  are  full  and  explicit  for  testing 
a  shunt-wound  motor.  Modifications  are  outlined  in  the  Ap- 
pendices for  applying  the  method  to  other  types  of  motors  and 
generators. 

§  2.  For  testing  any  machine  two  runs  are  made :  a  load  run 
to  ascertain  working  conditions,  and  a  no-load  run  (or  runs)  to 
determine  losses  under  these  same  conditions. 

In  making  the  no-load  run  for  losses  the  machine  can  be  driven 
electrically  as  a  motor  or  mechanically  as  a  generator.  The  for- 
mer method  is  used  in  this  experiment  (§  7)  ;  the  latter  method 
is  described  in  §  21  of  Appendix  I. 

The  resistance*  of  the  armature  is  to  be  found  by  the  fall  of 
potential  method  both  before  and  after  the  load  run,  in  order 
that  it  may  be  determined  both  cold  and  hot  (see  §  17,  Exp. 
i-A).  Since  this  includes  the  resistance  of  the  brushes  and  of 
brush  contact,  which  varies  with  current,  to  be  exact  it  would 
be  necessary  to  measure  the  armature  resistance  for  each  load 

their  reading.  Assume  the  true  output  to  be  95  when  the  true  input  is 
100.  The  output,  as  measured,  might  vary  from  94.05  to  95.95  and  the 
input,  from  99  to  101 ;  hence  the  efficiency,  determined  from  output, 
might  vary  from  93.1  to  96.9  per  cent.  On  the  other  hand  with  the  same 
percentage  error  in  their  determination,  the  measured  losses  might  vary 
from  4.95  to  5.05  and  the  measured  input  from  99  to  101 ;  hence  the 
efficiency,  determined  from  losses,  could  only  vary  from  94.9  per  cent,  to 
95.1  per  cent. 

*  In  measuring  armature  resistance  the  voltmeter  is  to  be  connected  to 
the  same  points  as  in  the  load  run. 


2-B]  EFFICIENCY.  43 

current.  No  account  will  be  taken  of  a  possible  difference  be- 
tween the  contact  resistance  with  machine  running  and  that 
measured  with  armature  stationary. 

§  3.  Load  Run  (Shunt  Motor). — This  run  is  made*  to  ascer- 
tain the  working  conditions  for  which  tbe  losses  are  to  be  deter- 
mined, that  is,  to  ascertain  the  load  current  and  hot  resistances 
for  calculating  copper  losses  and  to  ascertain  the  normal  speed 
and  excitation  for  which  the  iron  and  friction  losses  are  to  be 
determined  in  the  no-load  run.  (The  load  run  is  a  repetition 
of  the  run  made  in  Exp.  2-A  for  obtaining  speed  characteristics.) 

§  4.  Connect  the  motor  to  the  supply  lines,  the  voltage  of 
which  should  remain  practically  constant  during  the  run.  (See 
Fig.  i  of  Exp.  2-A.)  Adjust  the  field  current  by  means  of  the 
field  rheostat  so  that  the  motor  runs  at  its  rated  full-loadf  speed 
(or  the  speed  for  which  its  efficiency  is  desired)  and  keep  the 
field  current  constant  at  this  value  during  the  run.  Care  in 
keeping  the  field  current  constant  will  increase  the  accuracy  of 
the  results ;  it  is  not  sufficient  to  leave  the  rheostat  in  one  posi- 
tion and  assume  the  field  current  constant  because  it  is  very 
nearly  so. 

*  (§3a).  Omission  of  Load  Run. — It  will  be  seen  that  the  load  run  is 
not  essential  and  that  the  method  may  be  employed  even  when  the  load 
run  is  impossible.  Whenever  it  is  possible,  however,  the  load  run  should 
be  taken,  since  it  serves  to  get  the  machine  "  down  to  its  bearings,"  that  is, 
down  to  its  working  condition  of  friction  as  well  as  of  temperature. 

When  the  load  run  is  omitted,  cold  resistances  are  measured  and  hot 
resistances  determined  by  suitable  temperature  corrections  or  assumptions. 
Values  of  field  current  and  speed  are  determined  for  no  load;  values  are 
assumed  for  full  load  which  it  is  believed  will  most  nearly  represent  the 
operating  conditions  for  which  the  efficiency  is  to  be  obtained.  In  a 
motor,  for  example,  we  may  assume  a  constant  excitation  and  a  constant 
speed,  or  a  speed  which  is  say  5  per  cent,  lower  at  full  -load,  etc.  In  n 
generator  we  may  assume  a  constant  speed  and  a  constant  excitation,  or 
an  excitation  which  is  a  certain  amount  lower  (shunt  generator)  or  higher 
(compound  generator)  at  full  load. 

t(§4a).  For  commercial  testing  the  speed  should  be  adjusted  to  its 
rated  value  at  full  load;  in  laboratory  practice  the  adjustment,  when  de- 
sired, may  be  made  at  no  load. 


44  DIRECT   CURRENT   MOTORS.  [Exp. 

§  5.  Beginning  at  about  25  per  cent,  overload,  as  estimated  from 
the  input,  vary  the  load  by  steps  from  overload  to  no  load  or 
vice  versa;  at  each  step  measure  the  line  voltage,  armature  cur- 
rent,* field  current  and  speed. 

§  6.  The  motor  may  be  loaded  in  any  manner  that  is  convenient. 
A  brake  may  be  used  for  this,  but  it  is  frequently  more  con- 
venient to  load  with  a  generator  and  to  absorbf  the  output  of 
the  generator  by  suitable  resistances. 

§  7.  No-load  Run  (Shunt  Motor) ;  Machine  Driven  Electric- 
ally 4 — For  a  shunt  machine  one||  no-load  run  is  made;  the 
machine  is  operated  as  a  motor  at  the  same  constant  excitation 
as  in  the  load  run.  The  object  is  to  determine  the  losses  for 
different  speeds  at  this  constant  excitation.  Before  taking  read- 
ings the  motor  should  be  run  awhile  so  as  to  attain  its  normal 
working  condition  of  lubrication,  temperature,  etc. 

With  the  motor  running  unloaded,  adjust  the  field  current  to 
the  same  value  as  during  the  load  run  and  hold  constant  at  this 
value  during  the  no-load  run.  By  varying  the  electromotive 
force  impressed  on  the  armature  terminals,  vary  the  speed  of  the 
motor  by  steps  so  as  to  cover  as  wide  a  range  of  speed  as  pos- 
sible; this  will  give  more  accurate  results  than  if  only  the  speed 
range  of  the  load  run  is  covered.  At  each  step  measure  the 

*  See  §  i6c,  Exp.  2-A. 

flf  a  direct  current  generator  of  suitable  voltage  is  used,  the  current 
from  the  generator  may  be  "  pumped  back "  into  the  motor  supply  line 
(§26). 

$  (§7a).  This  run  can  be  made  with  the  machine  driven  mechanically 
(§21)  instead  of  electrically. 

||  (§7b).  Although  a  run  at  only  one  excitation  is  necessary  for  de- 
termining the  efficiency  of  a  shunt  motor,  runs  at  other  excitations  are 
recommended.  These  additional  runs  may  be  taken  by  the  two  voltage 
method  (§7d).  They  are  necessary  if  hysteresis  loss  is  to  be  separated 
(Appendix  I.)  or  if  flux  density  is  variable  (Appendix  III.).  If  a  run 
is  wanted  at  a  very  high  saturation,  a  higher  voltage  may  be  supplied  to 
the  field  than  the  rated  voltage  supplied  to  the  armature. 


2-B] 


EFFICIENCY. 


45 


Constant  Potential  Supply 


electromotive  force  impressed  on  the  armature  terminals,  arma- 
ture current,*  field  current  and  speed. 

By  using  two  resistances,  B  and  C,  arranged  as  in  Fig.  I,  the 
electromotive  force  impressed 
on  the  armature  may  be  varied 
by  short  circuiting  more  or 
less  of  B  or  of  C.  A  single 
series  resistance  B  may  suffice, 
but  the  adjustment  in  many 
cases  can  be  better  made  with 
two.  An  independent  genera- 
tor can  be  used  as  a  supply 
to  obtain  variable  voltages  for 
the  armature  circuit,  or  the 


\\ 


/^~~X  SHUNT  FIELD  X        X 

— '  *  j  j-nnRnnsTOP — •(  A/~- 


STARTING  BOX 


FIG.  i.     Connection  for  no-load  run  as 
a  shunt  motor  for  determining  losses. 


two    voltagesf    of    a    three- 
wire  system. 

§  8.  Results.— The     losses 
of  the  motor  include: 

(1)  Copper  losses  of  field 
and  armature; 

(2)  Iron  losses  of  armature; 

(3)  Friction  and  Windage,  or  air  resistance. 

Losses  (2)  and  (3)  are  rotation  losses  and  are  independent  of 
load. 

*  (§7c).  For  the  no  load  run  the  armature  current  is  small;  if  a  low 
reading  ammeter  is  used,  it  should  be  short-circuited  at  starting  to  avoid 
damage  by  the  initial  rush  of  starting  current. 

t(§7d).  Two-voltage  Method. — For  instruction  purposes  a  complete 
series  of  armature  voltages  and  corresponding  speeds  is  desirable.  Where 
two  supply  voltages  (as  no  and  220  volts  on  a  3-wire  system)  are  avail- 
able, accurate  results  may  be  obtained  by  a  two-voltage  method,  by  taking 
8  or  10  readings  and  averaging  first  with  say  220  and  then  with  no  volts 
impressed  on  the  armature  of  a  220  volt  motor.  These  points,  accurately 
determined,  are  sufficient  for  working  up  results  by  the  straight  line 
method  of  Fig.  2,  in  which  they  are  represented  by  black  dots  p  and  q. 
By  this  method  the  trouble  of  adjusting  armature  voltage  is  avoided. 


46  DIRECT    CURRENT    MOTORS.  [Exp. 

§  9.  Copper  Losses. — The  copper  losses  for  any  circuit  can  be 
computed,  if  the  current  and  resistance  through  which  it  flows 
are  known,  being  equal  to  RP  where  R  is  resistance  and  /  is  cur- 
rent. The  armature  copper  loss  is  thus  computed ;  it  is  a  vari- 
able loss,  changing  with  load. 

The  field  copper  loss  is  a  constant  loss  and  does  not  vary  with 
load.  It  also  can  be  computed  by  the  formula  RP,  or  more  con- 
veniently from  the  formula  El,  the  product  of  current  in  the  field 
circuit  and  voltage  supplied  at  its  terminals.  (The  formula  El 
cannot  be  thus  used  unless  copper  loss  is  the  only  expenditure 
of  energy;  it  cannot  be  used  for  determining  copper  loss  of  an 
armature  or  other  circuit  in  which  there  is  a  back  electromotive 
force.) 

In  a  self-excited  machine,  in  which  a  field  rheostat  is  used 
under  normal  operation,  the  loss  in  the  rheostat  is  to  be  included 
in  the  field  circuit  loss. 

§  10.  Iron  Losses. — The  iron  losses  are  losses  due  to  hysteresis 
and  eddy  currents  ;*  they  are  independent  of  load,  but  vary  with 
the  speed  and  with  the  flux  density  in  the  armature.  At  con- 
stant speed,  hysteresis  loss  (within  the  usual  working  range) 
varies  approximately  as  the  1.6  power  of  the  flux  density;  eddy 
currents  as  the  square  of  the  flux  density.  At  constant  flux 
density,  hysteresis  loss  varies  directly  with  the  speed  and  eddy 
currents  with  the  square  of  the  speed.  If  the  field  current  of 
the  motor  is  held  constant,  the  flux  density  in  the  armature  will 
be  practically  constant  for  all  loads.  It  will  be  modified  under 
loadf  to  a  small  extent  by  armature  reaction,  the  effect  of  which 
will  be  neglected.  Hence  in  a  shunt  motor  run  with  constant 

*  This  includes  eddy  currents  in  the  pole  pieces  and  in  armature  copper 
as  well  as  in  armature  iron. 

t  (§  loa).  Load  Losses. — Losses  which  occur  under  load  in  addition 
to  copper  losses  and  to  the  no-load  iron,  friction  and  windage  losses  are 
termed  load  losses.  Any  loss  due  to  field  distortion  constitutes  such  a 
loss.  Load  loses  are  usually  neglected  as  small  or  are  estimated.  See 
Standardization  Rules  114-7. 


2_B]  EFFICIENCY.  47 

field  current,  the  iron  losses  are  independent  of  load  and  depend 
upon  speed  alone. 

§  ii.  Friction  and  Windage. — The  friction  and  windage  losses 
are  also  independent  of  load  and  depend  alone  upon  speed,  being 
(for  all  practical  purposes)  directly  proportional*  to  speed. 
Friction  includes  frictions  of  brushes  as  well  as  of  bearings. 

§  12.  Rotation  Losses  W0  (Combined  Iron  Losses,  Friction 
and  Windage). — In  the  no-load  run  the  power  supplied  to  the 
armature  (product  of  armature  voltage  and  current)  gives  the 
rotation  losses  plus  a  small  armature  copper  loss.  This  copper 
loss  is  subtracted  (or  neglected  as  small)  to  get  the  rotation 
losses.  These  losses  are  sometimes  termed  stray  power.\ 

The  combined  rotation  losses  W0,  thus  determined  at  no  load, 
will  be  present  at  all  loads  and  will  have  the  same  value  for  the 
same  speed  and  excitation.  If  the  speed  of  the  motor  is  very 
nearly  constant,  the  W0  losses  will  be  correspondingly  constant. 
Rotation  losses  are  commonly  classed  among  the  constant  losses,! 
inasmuch  as  they  are  independent  of  load  and  the  variation  due 
to  any  small  change  of  speed  is  small. 

For  determining  efficiency  there  is  no  necessity  for  ascertain- 
ing the  separate  losses  due  to  hysteresis,  eddy  currents,  friction 
and  windage,  their  combined  value  W0  being  sufficient. 

§  13.  A  curve  should  be  plotted  showing  the  rotation  losses 
W0  for  constant  field  current  at  different  speeds.  To  plot  this 
curve  accurately,  it  is  best||  to  first  plot  for  various  speeds  the 

*(§na).  Windage  increases  more  rapidly  than  the  first  power  of  the 
speed ;  but  windage  loss  is  comparatively  small  and  does  not,  at  usual 
speeds,  materially  affect  the  law  of  variation  of  the  combined  friction  and 
windage  losses. 

t  The  term  stray  power  applies  to  any  loss  except  copper  loss. 

$(§i2a).  The  no-load  losses  are  the  rotation  losses  plus  the  copper 
loss  of  the  field  circuit  (and  the  practically  negligible  copper  loss  of  the 
armature)  ;  the  no-load  losses  are  therefore  termed  "constant." 

II  (§I3a)-  This  is  advantageous  because  a  straight  line  can  be  drawn 
more  accurately  than  a  curved  one,  when  the  observed  data  are  few 
or  irregular;  two  accurate  points  are  sufficient,  but  three  are  better  as  a 


48 


DIRECT   CURRENT   MOTORS. 


[Exp. 


values  of  W0  -=-  speed,  which  will  give  the  straight  line  ac  in 
Fig".  2.  At  very  low  speeds,  there  may  be  a  deviation  from  a 
straight  line,  due  possibly  to  errors  in  assumptions  as  to  friction, 
etc.,  at  these  speeds.  This,  however,  does  not  affect  the  accuracy 
of  the  construction;  the  straight  part  of  the  curve  is  to  be  ex- 
tended back  to  a. 


Ut 

s 


o 
cc  a 


SPEED    (R.R  M.) 

FIG.  2.  Variation  of  rotation  losses  W0  (iron  losses,  friction  and  windage) 
with  speed,  at  constant  field  excitation.  The  torque  dc  to  overcome  rotation 
losses  is  composed  of  db — to  overcome  friction,  windage  and  hysteresis — and 
be,  to  overcome  eddy  current  loss. 

After  plotting  this  line  for  W0  -r-  speed,  pick  off  values  from 
it  and  multiply  by  speed,  thus  getting  as  many*  points  as  desired 
for  plotting  the  W0  curve.  For  fuller  treatment,  see  Appendix  I. 

§  14.  Efficiency. — For  any  load  (corresponding  to  readings  in 
the  load  run,  or  assumed),  the  input  is  equal  to  the  product  of 
line  current  and  voltage. 

The  losses  are :  the  (variable)  RP  loss  in  the  armature  for 
the  particular  armature  current  7;  the  (constant)  copper  loss  of 

check.  It  is  always  desirable  to  plot  the  results  of  any  experiment,  if 
possible,  as  a  straight  line,  arc  of  circle,  or  as  some  curve  whose  law  is 
known.  The  arc  of  a  circle  is  much  used  in  alternating  current  testing. 
*  Obtained  in  this  way,  more  points  may  be  used  in  plotting  IV o  than 
the  number  of  observations. 


2-B] 


EFFICIENCY. 


49 


the  field;  and  the  (almost  constant)  rotation  loss  W0,  obtained 
from  the  curve  in  Fig.  2  for  the  particular  speed  and  excitation. 

The  output  is  found  by  subtracting  these  losses  from  the  in- 
put; the  efficiency  is  output  divided  by  input. 

§  15.  Curves  should  be  plotted  with  power  output  (or  more 
simply  with  armature  current)  as  abscissae,  showing  separate  and 
total  losses,  input,  output,  efficiency,  total  current  and  speed ;  also 
useful  torque  (watts  output  -f-  speed)  ;  see  Fig.  3.  Compare  the 
curves  of  Fig.  3  with  the  curves  for  a  transformer,  Fig.  4,  Exp. 
5~A,  and  Fig.  8,  Exp.  5-6. 


Field  RI2Los9 


POWER  OUTPUT 
FIG.  3.     Losses  and  efficiency  of  a  shunt  motor. 

Maximum  efficiency  occurs  when  the  variable  loss  (armature 
RI2)  equals  the  constant  losses;  see  §  28. 

It  is  seen  that  efficiency  at  light  loads  is  low;  this  is  true  of 
both  generators  and  motors.  For  this  reason  several  generators 
are  commonly  run  in  parallel  in  a  central  station ;  as  the  load  on 
the  station  decreases,  the  generators  are  cut  out  one  at  a  time, 
so  that  the  remaining  generators  will  be  more  or  less  fully  loaded 
and  will  run  nearer  the  point  of  maximum  efficiency. 


5°  DIRECT   CURRENT   MOTORS.  [Exp. 

APPENDIX   I. 

INTERPRETATION  OF  METHOD;  AND  SEPARATION  OF  LOSSES. 

§  1  6.  Interpretation*  of  Figure  2.  —  For  constant  flux  density  (con- 
stant field  excitation  in  a  shunt  machine),  the  losses  due  to  'hystere- 
sis, friction  and  windage  are  proportional  to  speed  (§§  10,  n)  and 
may  be  expressed  as  AS,  where  A  is  some  constant  and  6*  is  speed. 
Eddy  current  loss  being  proportional  to  the  square  of  the  speed  may 
be  expressed  as  BSZ,  in  which  B  is  some  constant.  The  total  rota- 
tion loss  is  accordingly  the  sum 


which  is  the  equation  of  the  WQ  curve  in  Fig.  2.     Dividing  by  S, 
we  have  the  torque  to  overcome  rotation  losses 


which  is  the  equation  of  the  straight  linef  ac  in  Fig.  2.  (See  §  3b, 
Exp.  2-  A.)  Extending  this  line  back  to  zero  speed  at  a  and  draw- 
ing the  horizontal  ab,  we  have  be  the  torque  to  overcome  eddy  cur- 
rent loss  (proportional  to  speed)  and  db  the  torque  to  overcome  hys- 
teresis, friction  and  windage  (independent  of  speed).  These  state- 
ments and  the  statements  made  in  the  following  paragraphs,  hold 
true  throughout  the  range  of  speeds  for  which  W0-^-S  is  a  straight 
line,  which  is  much  more  than  the  working  range  of  the  machine. 
§  17.  Determination  of  Watts  Eddy  Current  Loss.—  For  any  speed, 

*(§i6a).  The  principle  of  the  graphical  method  which  is  here  used 
was  brought  out  by  R.  H.  Housman  and  by  G.  Kapp,  independently,  in 
1891  (London  Electrician,  Vol.  XXVI.,  pp.  699  and  700)  ;  each  made  use 
of  a  straight  line  relation  for  plotting  data  obtained  by  running  a  motor 
at  constant  excitation  and  varying  armature  voltage.  The  details,  as  here 
given,  have  been  modified  by  the  writer  with  a  view  to  making  the  method 
simpler  and  more  useful.  The  original  papers  are  excellent,  but  their 
method  has  been  made  unnecessarily  cumbersome  by  writers  who  have 
followed  them.  Earlier,  Mordey  had  used  equations  similar  to  those  of 
§  16  for  analytical  separation  of  losses. 

t(§i6b).  Since,  at  constant  excitation,  armature  voltage  (or  more 
strictly  counter-electromotive  force)  is  proportional  to  speed,  the  Wo 
curve  can  be  drawn  with  E'  as  abscissae  instead  of  speed.  We  then 
divide  by  E'  (instead  of  S}  and  get  the  straight  line  ac,  the  ordinates  of 
which  (Wa  -:-£')  are  amperes. 


2-B]  EFFICIENCY.  51 

(multiplying  be  by  S,  gives  watts  eddy  current  loss;  multiplying  db 
by  5  gives  watts  loss  in  hysteresis,  friction  and  windage. 
If  the  eddy  current  loss  were  zero,  ac  would  coincide  with  the 
horizontal  line  ab;  the  first  equation  in  §  16  would  become  W0  =  AS, 
showing  that  W0  would  be  proportional  to  speed  and  the  W0  curve 
in  Fig.  2  would  become  a  straight  line. 

§  1 8.  A  Convenient  Approximation. — Since  the  eddy  current  loss 
is  commonly  only  a  small  part  of  the  total  rotation  losses,  for  small 
changes  in  speed  it  is  nearly  correct  and  often  very  convenient  to 
say  the  rotation  losses  IV0  are  directly  proportional  to  speed. 

§  19.  Further  Separation  of  Losses. — Hysteresis  loss  can  be  ap- 
proximately separated  from  friction  and  windage  by  additional  runs 
at  other  field  excitations.  Friction  and  windage  can  not  be  separated 
from  each  other  by  any  simple  means  and  hence  are  considered 
together.  There  are  various  graphical  and  analytical  methods  for 
separating  losses,  all  based  on  the  following  facts :  friction  and 
windage  losses  vary  as  first  power  of  speed  and  are  independent  of 
flux  density;  eddy  current  loss  varies  as  square  of  speed  and  square 
of  flux  density ;  hysteresis  loss  varies  as  first  power  of  speed  and 
1.6  power  of  flux  density.  At  any  one  speed,  armature  voltage  is 
taken  as  a  measure  of  flux  density.  In  any  of  these  methods  it  is 
necessary  to  make  some  assumption  or  approximation ;  for  this  rea- 
son the  graphical  methods  are  superior.  (In  the  graphical  method 
given  below  the  approximation  consists  in  obtaining  Oa0  by  extra- 
polation to  zero  excitation.) 

The  analytical  methods  will  not  be  taken  up  here;  they  consist  in 
obtaining  several  equations  (based  upon  the  above  relations)  and 
eliminating  between  them  after  substituting  numerical  values  obtaine'd 
from  observation  of  W0  at  various  speeds  and  flux  densities. 

§  20.  Graphical  Method. — Various  graphical  methods  for  separat- 
ing losses  differ  chiefly  in  detail;  the  following  procedure  (either 
a  or  b)  is  suggested: 

(a)  Make  a  series  of  no-load  runs,  as  already  described,  at  vari- 
ous field  excitations,  extending  these  to  as  low  a  field  excitation  as 
possible.  Plot  results  as  in  Fig.  2,  obtaining  a  series  of  curves 
(straight  lines)  ac  with  intercepts  Oa^  Oa2,  Oa3,  etc.,  corresponding 
to  various  field  currents.  It  is  desired  to  find  a  value  for  an  inter- 
cept Oa0  for  the  supposed  case  of  zero  field  current,  for  which  of 


52  DIRECT   CURRENT   MOTORS.  [Exp. 

course  no  run  can  be  made.  To  obtain  this,  plot  a  curve  showing 
Oalt  Oa2,  Oa3,  etc.,  for  various  field  currents  and  continue  the  curve 
back  to  zero  field  current  so  as  to  get  a  value  for  Oa0  by  extra- 
polation. 

(b)  It  will  be  found  by  experience  that  the  value  of  Oa  found  by 
a  run  at  a  very  low  field  excitation  will  differ  but  little  from  the 
desired  value  Oa0  for  zero  excitation;  that  is,  the  iron  losses  at  very 
low  excitation  are  negligible.  Instead  of  a  series  of  no-load  runs 
and  extrapolation,  one  no-load  run  is  taken  at  as  low  an  excitation 
as  possible ;  the  value  Oa  obtained  from  this  run  is  taken  as  the  value 
of  Oo0  which  would  be  obtained  at  zero  excitation. 

Referring  to  Fig.  2,  Oa0  obtained  by  either  procedure  just  described 
is  the  torque  to  overcome  friction  and  windage,  for  at  zero  excitation 
there  is  no  hysteresis  loss.  To  obtain  watts  loss  in  friction  and 
windage  at  any  speed,  multiply  Oa0  by  S;  this  is  independent  of 
excitation.  To  obtain  watts  loss  in  hysteresis  at  any  speed  for  some 
particular  excitation,  multiply  a0a  (for  that  excitation)  by  S. 

§21.  Determination  and  Separation  of  Losses;  Machine  Driven 
Mechanically  by  an  Auxiliary  Driving  Motor. — This  method,  with  the 
machine  driven  mechanically,  is  not  limited  to  testing  direct  current 
machines;  it  can  be  used  in  testing  alternators,  synchronous  motors, 
etc.  By  this  method  separate  values  are  found  for  the  iron  losses  and 
for  the  mechanical  losses;  that  is,  for  hysteresis  and  eddy  currents 
combined  and  for  friction  and  windage  combined. 

The  preceding  method,  with  the  machine  driven  electrically  (§7), 
gave  directly  the  eddy  current  loss  and  the  combined  hysteresis,  fric- 
tion and  windage  (§17).  Each  method  has  its  advantages;  in  the 
one  hysteresis  is  combined  with  eddy  current,  in  the  other  with  fric- 
tion and  windage. 

The  procedure  is  as  follows:  (i)  The  machine  to  be  tested  is  sepa- 
rately excited  and  is  driven  as  a  generator*  on  no  load  at  normal 
speed  and  excitation  by  means  of  a  shunt  motor ;  compare  §  25.  The 
motor  input  is  measured.  (2)  The  generator  field  circuit  is  broken 
and  motor  input  again  measured;  the  diminutionf  in  motor  input 

*  The  armature  winding  is  idle;  this  test  therefore  can  be  made  for 
finding  iron  loss  and  friction  of  a  machine  with  armature  unwound. 

t(§2ia).  This  assumes  that  the  motor  losses  remain  constant.  The 
small  change  in  armature  RI~  loss  will  usually  be  negligible;  if  not 


2-B]  EFFICIENCY.  53 

gives  the  iron  losses  (hysteresis  and  eddy  current)  of  the  generator. 
(3)  The  brushes  of  the  generator  are  lifted,  the  diminution  in  motor 
input  giving  brush  friction.  (4)  The  belt  is  next  thrown  off,  the 
diminution  in  motor  input  now  giving  the  generator  journal  friction, 
windage  and  the  belt  loss. 

The  iron  losses  may  be  found  for  various  excitations  at  normal 
speed.  These  losses  should  be  determined  for  an  increasing  excita- 
tion ;  the  losses  with  a  decreasing  excitation  would  be  more. 

For  obtaining  iron  losses  alone,  this  method  with  the  machine 
driven  mechanically  is  better  than  the  method  (§7)  with  the  ma- 
chine driven  electrically;  for  it  gives  iron  losses  directly,  separate 
from  friction,  and  it  is  not  necessary  to  go  through  any  separation 
of  losses  as  in  §  20.  This  avoids  error  due  to  extrapolation  and 
makes  no  assumption  that  friction  and  windage  are  directly  propor- 
tional to  speed. 

On  account  of  belt  tension,  journal  friction  will  be  more  than  in 
the  no-load  test  with  the  belt  off  (§7).  Belt  losses  are  also  included 
with  friction  and  windage.  This  may  sometimes  be  desirable,  since 
it  is  the  usual  condition  of  operation.  In  a  test  of  the  motor  per  se, 
these  losses  ought  not  to  be  included,  but  they  cannot  be  simply  sepa- 
rated (§  24a). 

If  the  loss  found  by  lifting  the  brushes  is  more  when  the  machine 
is  excited  than  when  not  excited,  the  brushes  are  not  in  the  neutral 
position,  thus  causing  additional  loss  by  current  circulating  through 
an  armature  coil  and  brush. 

If  it  is  desired  to  separate  the  iron  losses  into  components,  hyster- 
esis loss  and  eddy  current  loss,  runs  are  made  with  varying  speed  and 
a  constant  excitation  for  each  run.  For  each  run  plot  iron-loss  -=-  5" 
as  a  straight  line,  similar  to  ac  in  Fig.  2.-  For  any  speed,  the 
product  be  X  S  gives  watts  eddy  current  loss  for  the  particular  ex- 
citation ;  db  X  •$"  gives  watts  hysteresis  loss. 

negligible,  it  should  be  taken  into  account.  Belt  loss  cancels  out  and 
does  not  enter  into  the  determination  of  iron  losses  or  brush  loss. 


54  DIRECT   CURRENT   MOTORS.  [Exp. 

APPENDIX    II. 

MISCELLANEOUS  NOTES. 

§  22.  Efficiency  of  a  Generator. — To  find  the  efficiency  of  a  machine 
as  a  generator,  a  load  run  is  made  as  a  generator  to  ascertain  the 
working  conditions  of  speed,  excitation  and  voltage.  A  no-load  run 
as  a  motor  is  then  made  under  these  same  conditions.  The  load  run 
should  be  made  whenever  possible,  but  it  can  be  omitted  (§  3a).  In 
the  load  run,  the  field  rheostat  may  be  kept  in  one  position  (§  12, 
Exp.  i-B)  or  changed  so  as  to  maintain  the  desired  terminal  voltage 
(§26,  Exp.  i-B),  according  to  what  may  be  taken  as  the  working 
conditions  of  the  machine.  Commercially  the  latter  is  more  usual. 
For  a  compound  generator,  see  §  31. 

§  23.  Efficiency  of  a  Motor  Generator.* — A  load  run  is  to  be  made 
when  possible  and  measurements  made  of  the  various  currents  and 
voltages  for  both  motor  and  generator.  (See  §  3a.)  A  no-load  run 
is  to  be  made  if  possible  with  the  generator  uncoupled;  this  deter- 
mines the  motor  losses.  Next  make  a  run  with  the  generator 
coupled  but  not  excited,  the  increase  in  losses  over  the  no-load  run 
showing  the  friction  and  windage  of  the  generator.  Follow  this  with 
a  run  in  which  the  generator  has  its  proper  excitation,  the  increase 
in  losses  over  the  preceding  run  showing  the  iron  losses  of  the  gen- 
erator after  copper  losses  have  been  taken  into  account.  This  last 
run  gives  the  combined  rotation  losses  for  both  machines.  The  cop- 
per losses  are  computed  and  added  to  these  to  get  the  total  losses; 
knowing  these,  the  efficiencies  are  readily  computed  for  the  two 
machines,  combined  and  separately.  As  in  the  case  of  a  generator 
or  motor,  due  care  is  to  be  taken  in  all  the  no-load  runs  to  have  the 
proper  speed  and  flux  density  in  both  machines.  If  the  flux  density 
in  either  machine  was  not  constant  in  all  the  runs  (as  would  be  the 
case  in  a  compound  or  differential  machine),  take  note  of  Appendix 
III.  The  test  may  be  made  by  reversing  the  set,  that  is,  running  the 
generator  as  a  motor ;  this  makes  it  possible  to  determine  the  friction 
and  windage  of  the  motor  separate  from  iron  losses. 

§  24.  Calibrated  Generator  for  Measuring  Motor  Output. — The  out- 

*  The  details  of  this  test  can  be  modified  according  to  circumstance; 
see  §21. 


2-B]  EFFICIENCY.  55 

put  of  a  motor  can  be  determined  if  for  a  load  it  drives  a  shunt  gen- 
erator whose  losses  are  known;  it  is  best  to  have  the  generator 
separately  excited. 

The  motor  output  is  equal  to  the  power  taken  to  drive  the  genera- 
tor, that  is,  to  the  measured  generator  output  (El}  plus  generator 
losses.  The  losses  are  the  copper  losses  and  the  rotation  losses 
picked  from  curves  (as  in  Fig.  2)  for  the  particular  speed  and  exci- 
tation ;  to  this  should  be  added  the  belt  losses,* — a  small  but  uncer- 
tain quantity.  If  the  generator  is  separately  excited,  no  account  need 
be  taken  of  field  copper  loss. 

§  25.  Calibrated  Motor  for  Measuring  Power  to  Drive  a  Generator. 
—The  power  used  in  driving  a  generator  can  be  determined  if  it  be 
driven  by  a  shunt  motor  whose  losses  are  known.  The  power  taken 
to  drive  the  generator  is  equal  to  the  motor  input  (El  for  the  arma- 
ture) less  armature  RI2,  less  W9  for  the  particular  speed  and  exci- 
tation, less  belt  loss  (§24a). 

§26.  Return  of  Power  to  Line  by  "Loading  Back." — If  a  direct 
current  generator  of  suitable  voltage  is  used  as  a  load  for  a  direct 
current  motor,  the  current  from  the  generator  may  be  "  pumped 
back"  into  the  motor  supply  line  (or  into  any  other  supply  line). 
Used  as  a  method  of  loading,  it  saves  power,  avoids  the  necessity  of 
providing  load  resistances  for  the  generator  and  introduces  little 
complication. 

The  variation  in  load  put  upon  the  motor  in  driving  the  generator 
is  obtained  by  varying  the  generator  field  current.  First  let  us  sup- 
pose that  this  is  adjusted  until  the  generator  generates  a  voltage 
equal  to  the  line  voltage.  When  connected  to  the  line  (the  positive 
terminal  to  the  positive  line),  the  generator  will  now  neither  give 
nor  receive  current,  that  is,  will  neither  give  power  to  nor  receive 
power  from  the  line.  (At  a  lower  excitation,  it  will  receive  power 
as  a  motor.)  If  the  field  current  of  the  generator  is  now  increased, 
it  will  generate  a  voltage  higher  than  that  of  the  line  and  will  supply 
power  to  the  line.  This  power  can  be  increased  by  a  further  increase 

*  (§243).  Belt  Losses. — Cotterill  (Applied  Mechanics,  p.  365)  says:  "In 
ordinary  belting  this  loss  is  small,  not  exceeding  2  per  cent."  The  belt, 
on  account  of  its  tension,  also  increases  the  journal  friction  of  both 
motor  and  generator. 


56  DIRECT    CURRENT   MOTORS.  [Exp. 

in  excitation,  thus  increasing  the  load  on  the  driving  motor  as 
desired. 

When  the  loading  back  method  is  thus  used  simply  as  a  loading 
method  and  not  as  a  testing  method  (§27),  no  measurements  are 
made  on  the  generator;  measurements  are  made  on  the  motor  the 
same  as  though  the  generator  were  loaded  with  resistances. 

Since  one  machine  takes  power  as  a  motor  and  the  other  returns 
it  as  a  generator,  the  net  power  taken  from  the  supply  line  is  only 
that  which  is  required  to  supply  the  losses  in  the  two  machines. 

§  27.  Opposition  Method  for  Testing  Two  Similar  Machines. — If 
two  similar  machines  are  operated  as  in  the  preceding  paragraph 
and  measurements  are  taken  on  both,  they  can  be  tested  by  Kapp's* 
opposition  method  and  their  combined  losses  determined. 

There  are  various  other  opposition  methods  for  accomplishing  the 
same  object;  in  each  of  these  two  similar  machines  are  run,  one  as 
motor  and  the  other  as  generator  under  load  conditions.  The  two 
machines  are  connected  both  electrically  and  mechanically,  so  that 
power  circulates  between  them  and  the  only  outside  power  taken  is 
that  necessary  to  supply  the  combined  losses.  These  losses  may  be 
all  supplied  by  the  line  (Kapp's  method)  or  either  partly  or  wholly 
by  an  auxiliary  motor  or  by  an  auxiliary  booster,  giving  rise  to  the 
various  methodsf  of  Hopkinson,  Potier,  Hutchinson  and  Blondel. 

Although  opposition  methods  are  economical  of  power,  they  are 
not  economical  of  time  or  apparatus;  they  are  accordingly  limited  to 
testing  pairs  of  large  machines  which  could  not  be  tested  under  load 
conditions  in  any  other  way.  Temperature  runs,  regulation  and 
efficiency  tests  are  made  in  this  way.  Kapp's  method  is  the  simplest, 
but  (on  account  of  the  different  field  excitation  of  the  two  machines) 
theoretically  is  not  so  accurate  as  some  of  the  other  methods. 

§  28.  Point  of  Maximum  Efficiency.— Consider  that  a  machine  has 
a  certain  constant  loss  (W9-\-  field  copper  loss)  and  a  variable  loss 
(armature  RI2)  which  varies  as  the  square  of  the  load  current  /  and 

*This  method  and  a  modification  by  Prof.  W.  L.  Puffer  is  fully  de- 
scribed in  Foster's  Electrical  Engineering  Pocketbook;  see  also  §  273. 

t(§27a).  For  full  description  and  complete  references,  see  Swenson 
and  Frankenfield's  Testing  of  Electromagnetic  Machinery;  see  also  R.  E. 
Workman,  Electric  Journal,  Vol.  L,  1904,  pp.  244,  289,  363;  Karapetoff's 
Exp.  Elect.  Eng.i  and  various  text  and  handbooks. 


2-B] 


EFFICIENCY. 


57 


hence  as  the  square  of  El  (the  line  voltage  E  being  constant). 
These  are  shown  in  Fig.  4,  in  which  the  curve  for  total  losses  is  a 
parabola. 

At  any  point  P  on  the  total  loss  curve,  the  loss  PA,  expressed  as 
a  percentage  of  El,  is  PA  -±-  OA,  which  is  the  tangent  of  the  angle 
POA.  It  is  clear  that  this  percentage 
loss  is  a  minimum  (and  the  efficiency 
a  maximum)  for  the  point  P'  where 
the  line  OP'  is  tangent  to  the  total 
loss  curve.  But  at  this  point  P',  we 
have  A'B'  =  B'P'.  (From  the  prop- 
erties of  a  parabola,  Od  is  bisected  at 
c.)  Hence: — For  any  apparatus  hav- 
ing a  constant  loss  and  a  variable  loss 
proportional  to  load  current,  maximum 
efficiency  occurs  at  such  a  load  that 
the  constant  loss  and  variable  loss 
are  equal.  The  same  result  can  be 
shown  analytically  by  obtaining  an 
expression  for  efficiency,  differentiat- 
ing and  equating  to  zero  (See  Franklin  and  Esty's  Electrical  Engi- 
neering, L,  137). 

This  is  true  for  any  apparatus;  thus,  in  a  transformer,  the  effi- 
ciency is  a  maximum  when  the  copper  loss  and  constant  core  loss  are 
equal.  Within  limits  the  designer  may  make  the  efficiency  a  maxi- 
mum at  the  particular  load  he  desires,  giving  due  consideration  to 
expense  and  to  the  uses  to  which  the  apparatus  is  to  be  put. 

APPENDIX    III.     , 

MODIFICATION  FOR  VARYING  FLUX  DENSITIES. 

§  29.  In  the  foregoing  tests,  the  load  run  was  made  with  constant 
field  excitation,  and  hence  at  constant  flux  density;  the  no-load  run 
was  made  at  this  same  constant  flux  density.  In  cases  where  the 
flux  density  varies  during  the  load  run  (due  to  a  variation  in  the 
shunt  field  current  or  due  to  the  action  of  the  series  field  coil  in  a 
compound,  differential  or  series  wound  machine),  three  (or  more) 
no-load  runs  should  be  made  at  three  different  flux  densities. 


0  A  El 

FIG.  4.  Total  losses  repre- 
sented by  a  parabola ;  P'  is 
point  of  maximum  efficiency. 


DIRECT    CURRENT   MOTORS. 


[Exp. 


FIG.  5. 


for  different 


The  following  is  suggested  as  a  method  for  conducting  the  test. 
§  30.  Varying  Excitation,  Shunt  Machine. — First  let  us  consider 
the  case  of  a  shunt  machine,  in  which  the  excitation  varied  during 
the  load  run.  Make  three  no-load  runs  at  three  excitations  covering 
the  range  of  excitations  used  in  the  load  run.  From  these  no-load 
runs,  after  plotting  W0-±-S,  plot  three  curves  (A,  B,  C  in  Fig.  5) 
showing  W0  for  different  speeds  as  before. 

To  get  W9  for  a  particular  speed,  erect  a  perpendicular  in  Fig.  5, 

corresponding  to  that  speed. 
This  perpendicular  intersects 
the  three  curves  A,  B,  C,  giv- 
ing (for  a  particular  speed) 
the  values  of  IV0  for  different 
field  currents.  For  each 
speed  a  derived  curve  may 
now  be  plotted  giving  W^ 
for  different  field  currents. 
§  31.  Compound  Generator. 
— In  testing  a  compound  gen- 
erator, first  make  a  load  run 
to  ascertain  the  equivalent  shunt  excitation  and  then  make  no-load 
runs  as  a  shunt  motor. 

Load  Run. — Make  a  load  run  as  a  compound  generator,  and  note 
the  values  of  terminal  voltage  and  speed  at  three  (or  more)  differ- 
ent loads ;  in  each  case  ascertain  the  equivalent  shunt  excitation,  i.  e., 
the  field  current  which  would  give  the  same  terminal  voltage  (and 
hence  the  same  flux  density)  with  the  machine  run  as  a  shunt*  gen- 
erator at  the  same  speed. 

No-load  Runs. — Knowing  this  equivalent  shunt  excitation,  make 
the  three  corresponding  no-load  runs  as  a  shunt  motor  at  constant 
excitation,  in  each  run  using  one  of  the  three  equivalent  shunt  field 
currents  just  determined. 

*(§3!a).  This  equivalent  shunt  excitation  may  be  determined  after 
each  reading:  without  stopping  the  machine,  the  series  winding  should 
be  first  short  circuited  and  then  opened ;  or,  the  machine  may  be  stopped 
and  started  again.  Instead  of  this  the  equivalent  excitation  can  be  found 
from  a  separate  shunt  run  (like  an  armature  characteristic  §26,  Exp.  i-B) 
in  which  is  determined  the  field  current  which  will  give  for  each  load 
the  same  terminal  voltage  as  in  tfie  compound  run. 


SPEED 

Rotation  losses 
excitations. 


2-B]  EFFICIENCY.  59 

Results. — Results  are  worked  up  as  in  the  preceding  paragraph. 
Curves  are  plotted  as  in  Fig.  5  and  derived  curves  found  showing 
the  variation  of  W^  with  field  current  for  any  speed.  Such  a  derived 
curve  is  plotted  for  each  speed  observed  in  the  load  run. 

§  32.  Compound  or  Differential  Motor. — A  load  run  is  first  made 
to  find  the  equivalent  shunt  excitation ;  no-load  runs  are  then  made 
as  a  shunt  motor. 

Load  Run. — Make  a  load  run  as  a  compound  or  differential  m'otor, 
and  note  the  speed  at  three  (or  more)  different  loads  so  chosen  as 
to  cover  the  speed  variation  of  the  run.  In  each  case  ascertain  the 
equivalent  shunt  excitation,  i.  e.,  the  field  current  which  would  give 
the  same  speed*  (and  hence  the  same  flux  density)  with  the  machine 
run  as  a  shunt  motor, — the  load  and  the  line  voltage  being  the  same 
as  before. 

No-load  Runs. — Knowing  this  equivalent  shunt  excitation,  make 
the  three  corresponding  no-load  runs  as  a  shunt  motor  at  constant 
excitation,  each  run  using  one  of  the  three  equivalent  shunt  field 
currents  just  determined. 

Results. — The  results  are  worked  up  as  in  the  preceding  para- 
graphs. From  the  three  no-load  runs  three  curves  are  plotted,  as 
in  Fig.  5,  showing  W \  for  varying  speed  at  different  excitations. 
From  these  curves  a  derived  curve  may  be  plotted  showing  the  varia- 
tion of  W0  with  field  excitation  for  any  speed.  Such  a  derived  curve 
is  plotted  for  each  speed  observed  in  the  load  run,  and  from  it  the 
value  of  W9  obtained  for  the  corresponding  excitation. 

*(§32a).  The  equivalent  shunt  excitation  may  be  determined  after 
each  reading  by  cutting  out  the  series  coil  as  in  §  3ia. 

The  adjustment  to  a  definite  speed  is,  however,  difficult  without  some 
particularly  sensitive  tachometer.  To  avoid  this  adjustment,  proceed  as 
follows : 

Determine  say  five  shunt  speed  characteristics,  that  is  make  five  runs 
at  different  constant  shunt  excitations,  determining  speed  for  different 
loads.  For  each  excitation  plot  speed  as  ordinates  and  armature  current 
as  abscissae.  By  interpolating  between  these  curves,  we  can  find  the  shunt 
excitation  that  gives  a  particular  speed  for  a  particular  armature  current. 
This  will  give  the  equivalent  shunt  excitation  corresponding  to  any  speed 
and  armature  current  found  in  the  load  run  as  a  differential  or  com- 
pound motor.  Knowing  the  equivalent  shunt  excitation,  the  correspond- 
ing no-load  runs  are  made. 


60  DIRECT    CURRENT    MOTORS.  [Exp. 

§  33.  Series  Motor. — A  series  motor  may  be  tested  for  losses  in 
substantially  the  same  manner  as  a.  shunt  motor.  So  far  as  losses 
are  concerned,  a  series  motor  is  like  a  shunt  motor  in  that  the  losses 
are  the  copper  losses,  which  can  be  computed,  and  the  rotation  losses 
W0  which  depend  only  upon  speed  and  excitation.  In  a  series  motor, 
however,  speed  and  excitation  vary  greatly  with  load. 

Load  Run. — A  load  run  as  a  series  motor  is  taken  to  obtain  speed 
and  current  for  different  loads;  see  Appendix  L,  Exp.  2-A.  (If  the 
no-load  run  is  to  be  taken  as  in  §  36,  the  load  run  is  not  a  necessity.) 

§34.  No-load  Runs  for  Obtaining  Rotation  Losses;  General  Pro- 
cedure.— No-load  runs  may  then  be  made  at  different  constant  exci- 
tations and  W^  found  for  different  speeds  by  varying  the  armature 
voltage  and  measuring  armature  input  in  the  usual  manner.  Read- 
ings are  taken  of  field  current,  armature  current  and  armature  volt- 
age. (The  procedure  is  sometimes  to  take  runs  with  constant  arma- 
ture voltage  and  varying  excitation.)  Any  convenient  means  may 
be  employed  for  obtaining  the  proper  constant  excitation  and  the 
desired  armature  voltage;  the  armature  and  field  can  best  be  sup- 
plied separately  and  not  in  series  (see  also  §37).  Curves  are  plotted 
for  each  excitation,  as  in  Fig.  5,  showing  W9  for  different  speeds. 
Instead  of  speeds,  armature  voltage  is  commonly  plotted  as  abscissae. 

§35.  No-load  Run  for  Obtaining  Rotation  Losses;  Special  Pro- 
cedure.— No-load  runs,  taken  as  in  §34,  gives  curves  (Fig.  5)  which 
tell  the  complete  story,  giving  rotation  losses  for  different  speeds  and 
field  currents.  As  a  matter  of  fact  such  complete  information  is 
often  unnecessary;  for,  with  constant  potential  supply,  a  series  motor 
has  a  definite  counter-electromotive  force  and  a  definite  speed  for 
any  particular  current  (see  Fig.  3  of  Exp.  2-A).  It  is  necessary, 
therefore,  to  get  the  rotation  losses  with  each  field  current  for  the 
one  corresponding  speed  only,  this  speed  being  obtained  by  supplying 
the  armature  with  the  proper  voltage. 

§  36.  This  proper  voltage  to  supply  the  armature  could  be  found 
by  trial  (being  adjusted  until  the  speed  in  the  no-load  run  for  a  par- 
ticular field  current  is  the  same  as  in  the  load  run  for  the  same 
current).  It  is  easier,  however,  to  compute  this  voltage  without 
making  a  load  run. 

We  know  that  in  any  run  (load  or  no  load)  speed  is  proportional 
to  counter-electromotive  force  for  the  same  excitation.  For  a  par- 


2-B]  EFFICIENCY.  61 

ticular  field  current  /  we  will,  therefore,  have  the  same  speed  in  the 
no-load  run  as  in  an  assumed  load  run  with  current  I,  if  in  the  no- 
load  run  the  counter-electromotive  force  (which  in  this  case  is  the 
impressed*  armature  voltage)  is  equal  to  the  counter-electromotive 
force  of  the  assumed  load  run.  But  for  the  load  run  we  can  com- 
pute the  counter-electromotive  force,  Er  =  E  —  RI,  for  any  assumed 
load  current  /.  (Here  E  is  the  rated  or  assumed  constant  line  volt- 
age for  which  the  losses  are  desired;  R  is  the  hot  resistance  of  the 
armature  and  field,  including  brushes,  etc.)  Hence  this  is  the  proper 
voltage  to  supply  the  armature  in  the  no-load  run  when  the  field 
current  is  /. 

In  testing  a  series  motor  by  this  method  the  field  is  excited  with 
current  I,  which  is  given  successive  values,  and  the  armature  is  sup- 
plied with  the  corresponding  proper  voltage,  E  —  RI.  (Or  the 
armature  can  be  given  successive  voltages  and  /  adjusted  to  cor- 
respond.) 

§  37.  A  convenient  method  sometimes  used  for  adjusting  field  cur- 
rent and  armature  voltage  to  their  proper  corresponding  values  is 
to  connect  the  field  and  armature  in  series  as  a  series  motor  with 
one  regulating  resistance  in  series  with  the  line  and  one  in  shunt 
with  the  armature.  For  the  first  reading  the  series  resistance  is 
adjusted;  after  that,  adjusting  the  shunt  resistance  alone  will  tend 
to  cause  the  field  current  and  armature  voltage  to  assume  automat- 
ically their  correct  relative  values.  (For  this  condition  the  series 
resistance  is  made  equal  to  the  armature  resistance.)  For  modified 
ways  of  conducting  the  test,  see  R.  E.  Workman,  Electric  Journal, 
L,  169. 

§  38.  No-load  Run  for  Friction. — When  the  field  current  is  very 
small,  hysteresis  and  eddy  current  losses  are  S£>  small  that  W0  gives 
practically  the  friction  and  windage  loss;  compare  paragraph  (&), 
§  20.  A  run  at  low  field  excitation  can  be  made  as  in  §  34.  This 
run,  however,  can  most  conveniently  be  made  with  the  field  and 
armature  in  series,  the  motor  being  run  as  a  series  motor  on  no  load 
at  a  low  voltage.  The  voltage  and  the  speed  are  controlled  by  a 
series  resistance;  no  shunt  resistance  is  used.  At  no  load  the  cur- 
rent through  the  field  is  so  small  that  iron  losses  in  the  armature 
are  negligible. 

*  The  copper  drop  due  to  armature  resistance  at  no  load  can  be  neglected 
or  a  small  correction  made. 


CHAPTER   III. 
SYNCHRONOUS   ALTERNATORS. 

EXPERIMENT  3-A.    Alternator  Characteristics.* 

§  i  Introductory. — Alternating  current  generators  are  usually 
synchronous.  Any  machine — generator,  motor  or  converter — 
is  said  to  be  synchronous  when  the  current  which  it  delivers  or 
receives  has  a  frequency  proportional  to  the  speed  of  the  ma- 
chine; otherwise  it  is  asynchronousf  or  non-synchronous. 

In  a  synchronous  machine,  the  current  or  electromotive  force 
has  one  half-wave  or  alternation — first  positive  and  then  nega- 
tive— for  each  pole  passed  by  a  given  armature  conductor.  A 
cycle  is  a  complete  wave  of  two  alternations.  In  a  synchronous 
machine,  there  is,  therefore,  one  cycle  for  each  pair  of  poles 
passed;  the  frequency  (cycles  per  second)  is,  accordingly,  equal 
to  the  speed  (in  revolutions  per  second)  multiplied  by  the  num- 
ber of  pairs  of  poles. 

To  deliver  current  with  a  frequency  of  60  cycles  per  second 
(7,200  alternations  per  minute),  a  bipolar  alternator  would  have 
to  be  driven  at  60  revolutions  per  second,  or  at  3,600  revolutions 
per  minute;  a  4-pole  machine,  at  1,800  revolutions  per  minute, 
etc.  Alternators  are  commonly  made  multipolar,  and  usually 
with  manyj  poles,  so  as  to  avoid  excessive  speed. 

*  The  curves  used  to  illustrate  this  experiment  and  Exp.  3-B  all  relate  to 
the  same  machine. 

t  (§  ia).  The  induction  motor  and  the  induction  generator  are  asyn- 
chronous. An  induction  motor  must  run  below  synchronous  speed,  i.  e., 
there  must  be  a  certain  slip,  in  order  to  produce  power.  An  induction 
generator,  on  the  other  hand,  must  be  driven  above  synchronous  speed  in 
order  to  generate  an  electromotive  force. 

$  (§  ib).  The  high  speed  of  the  steam  turbine  has  made  possible,  in  fact 
has  made  necessary,  large  alternators  with  only  few  poles ;  for  example,  a 
bipolar  10,000  K.  W.  turbo-alternator,  1,500  revolutions  per  minute,  is  men- 

62 


3-A1  CHARACTERISTICS.  63 

§  2.  Types  of  Alternators. — Synchronous  alternators  are  of 
the  following  three  types  (compare  Part  I.,  Exp.  i-A)  ; 

1.  Alternators   having   a   revolving   armature   and   stationary 
field,  used  only  for  small  machines. 

2.  Alternators  having  a  revolving  field  and  stationary  arma- 
ture, the  most  common  type. 

3.  Inductor  alternators,  having  a  stationary  armature  and  sta- 
tionary field,  the  revolving  part  or  inductor  consisting  only  of 
iron. 

The  first  type  corresponds  to  the  nearly  universal  type  of 
direct  current  generator;  there  is,  however,  no  commutator  and 
alternating  current  is  delivered  from  the  armature  winding  to 
the  line  by  means  of  collector  (or  slip)  rings  and  brushes.  In  the 
second  and  third  types,  the  armature  is  stationary  and  current  is 
delivered  directly  to  the  line  without  collector  rings.  The  con- 
tinuity in  insulation,  thus  made  possible,  is  an  important  ad- 
vantage in  high  potential  machines.  In  revolving  field  alterna- 
tors, the  field  current  is  introduced  through  slip  rings. 

Each  type  is  made  in  several  forms  which  may  be  studied  by 
reference  to  standard  works,*  or  better  by  examination  of  actual 
machines.  The  form  most  desirable  depends  upon  conditions 
of  operation,  character  and  speed  of  prime-mover,  etc.  In  some 
cases,  it  is  desirable  to  make  the  moving  mass  as  small  as  possi- 
ble ;  in  other  cases — as  in  direct-connected  engine-driven  genera- 
tors— a  certain  fly-wheel  effect  is  advantageous.  Alternators  of 
the  second  type  usually  have  an  internal  revolving  field,  a  con- 
spicuous exception  being  the  umbrella  form  of  external  revolving 
field  in  the  vertical-shaft  alternators  at  Niagara.  In  the  old 
Mordey  and  Brush  form  of  machine,  the  stationary  armature 
coils  were  in  a  vertical  plane  between  the  two  parts  of  the  revolv- 

tioned  in  Electric  Journal,  p.  550,  October,  1908.      The  steam  turbine  has  thus 
modified  both  alternating  and  direct  current  generators  (§3a,  Exp.  i-A). 
*  See  also  "  The  Mechanical   Construction  of  Revolving-field  Alterna- 
tors," by  D.  B.  Rushmore,  Transactions  A.  I.  E.  E.,  Vol.  XXIII.,  p.  253. 


64  SYNCHRONOUS   ALTERNATORS.  [Exp. 

ing  field.  The  inductor  alternator,  although  possessing  obvious 
mechanical  advantages,  is  handicapped  by  large  magnetic  leakage 
and  consequent  poor  regulation — unless  built  in  an  expensive 
manner  with  much  material. 

§  3.  Choice  of  Frequency. — In  the  early,  applications  of  alter- 
nating current,  when  power  transmission  was  not  developed  and 
current  was  used  for  lighting  only,  the  common  frequencies  in 
America  were  125  and  133^  cycles  per  second,  and  these  fre- 
quencies were  satisfactory  for  the  service.  The  efficiency  of  a 
transformer  increases  with  the  frequency  (Exp.  5-B),  and  from 
this  consideration  even  a  higher  frequency  would  be  desirable; 
but  as  frequency  is  increased,  we  have  greater  inductive  drop 
and  poorer  regulation  in  generator,  line  and  transformer. 

The  rotary  converter,  introduced  in  the  early  nineties,  required 
a  lower  frequency.  The  highest  frequency  at  which  it  can  operate 
is  practically  60  cycles,  and  25  cycles  is  better.  With  its  advent, 
the  higher  frequencies  were  abandoned ;  25  and  60  cycles  became 
standard,  the  former  for  power  alone,  and  the  latter  for  lighting 
and  (usually)  for  combined  power  and  lighting.  The  induction 
motor  has  its  best*  operation  within  this  same  range. 

Below  25  cycles,  or  thereabouts,  the  flicker  of  incandescent 
lamps  of  the  usual  types  becomes  prohibitive.  On  account  of  the 
high  speed  of  the  steam  turbine,  it  is  not  adapted  for  driving 
generators  below  25  cycles.  The  series  alternating  current 
motor,  which  is  more  economical  the  lower  the  frequency,  is  prac- 
tically the  only  apparatus  for  which  a  frequency  lower  than  25 
cycles  is  desirable.  As  the  art  progresses,  it  is  possible  that 
some  new  application  may  be  developed  which  will  demand  a 
frequency  much  higher  or  lower  than  the  frequencies  now  recog- 
nized as  standard. 

*  (§3a).  In  a  discussion  on  the  choice  of  frequency,  A.  I.  E.  E.,  Vol. 
XXVI.,  p.  1400,  June,  1907,  Dr.  Steinmetz  stated  that  the  most  efficient 
frequency  for  the  induction  motor  is  40  cycles,  the  best  frequency  for  small 
motors  being  higher  and  for  large  motors  lower.  He  also  states  that,  for 
converters,  25  cycles  is  better  than  either  a  higher  or  lower  frequency. 


3-AJ  CHARACTERISTICS.  65 

§4.  Characteristics. — Four  characteristics  are  to  be  taken: 

One  no-load  characteristic — the  no-load  saturation  curve. 

Three  load  characteristics — the  external  characteristic,  the  full- 
load  saturation  curve,  and  the  armature  characteristic. 

These  characteristics  are  similar  to  the  corresponding  char- 
acteristics of  a  direct  current  generator,  Exp.  i-B. 

§  5.  No-load  Saturation  Curve.* — This  curve  shows  the  ter- 
minal voltage  for  different  values  of  field  current,  the  machine 
being  driven  at  constant  speedf  without  load.     The  connections 
are  shown  in  Fig.  i.     Data  are  taken 
in  the  same  way  as  for  the  no-load       + 
saturation  curve  of  a  direct  current 
generator    (§§5-10,   Exp.   i-B)  ;   the 

alternator,     however,     is     necessarily      II"       L — .  ^ 
,  ,          .,    .  — Mwm-LJ 

separately  excited.  RHEOSTAT 

The  field  current  can  be  varied!  by      FlG*  '•    Connections  for  n°- 

load  saturation  curve. 

a  field  rheostat  in  series,  as  in  Fig.  i 

(with  a  second  rheostat  in  series  with  it  to  give  greater  range, 
if  necessary)  ;  or  by  an  arrangement  of  resistances  as  in  Figs. 
4  and  5,  of  Exp.  i-A. 

Voltage  readings  are  corrected  by  proportion  for  any  varia- 
tion of  speed,  and  plotted  as  in  Fig.  2.  Descending,  as  well  as 
ascending,  values  may  be  plotted  when  desired.  The  saturation 
factor  and  per  cent,  saturation  are  determined  as  in  Fig.  2,  of 
Exp.  i-B. 

§  6.  External  Characteristic. — This  curve-  shows  the  variation 
in  terminal  voltage  with  load.  The  alternator  is  driven  at  nor- 

*  (§5a).  If  the  alternator  is  motor-driven,  it  is  commercial  practice  to 
determine  its  core  loss  and  friction  at  the  same  time  that  the  no-load 
saturation  curve  is  taken.  See  §  15. 

f  (§5b).  Speed  and  frequency  are  proportional;  with  a  good  frequency 
meter  at  hand,  it  may  be  more  convenient  to  observe  frequency  than 
speed.  If  the  speed  can  be  varied,  note  that  voltage  is  proportional  to 
speed. 

$(§5c).  As  in  all   such  curves,  the  variation  should  be  made  continu- 
ously and  no  back  steps  should  be  taken  (§  7,  Exp.  i-A). 
6 


66 


SYNCHRONOUS  ALTERNATORS. 


[Exp. 


900 


mal  speed  and  excitation,  readings  being  taken  of  speed  and  field 

current  to  see  that  they  are  constant.    The  connections  are  shown 

in  Fig.  3. 

The  characteristic  is  to  be  obtained  for  unity*  power  factor,  a 

non-inductive  variable  resistance  being  used  for  a  load.     Read- 

ings are  taken  of  termi- 
nal voltage  and  external 
current  from  o  to  25  per 
cent,  overload. 

In  commercial  testing, 
the  excitation  is  adjusted 
for  normal  voltage  at 
full  load.  Fig.  4  shows 
the  characteristic  of  a 
25  K.W.  alternator  in 
which  the  voltage  in- 
creases from  575  at  full 
load  to  627  at  no  load  — 
a  regulation  of  9  per 
8  9  10  11  12  13  u  is  16  cent.  (See  §§14,  17, 
Exp.  i-B.)  It  is  de- 

Sirabl«   tO  hae  **     ^' 


o    l 


234567 

FIELD  AMPERES 


FIG.  2.  Saturation  curve  at  no  load,  and  at 
full  load  (43.4  amperes  at  unity  power  factor). 
Field  ampere-turns  equal  field  amperes  multi- 
plied  by  number  of  field  turns,  464. 

All  curves  in  Exps.  3-A  and  3-B  relate  to 

A.  ,  -,  i 

tne  same  25-kilowatt  alternator. 


ulation     as         close         as 

possible,    i.    €.,    with    the 
,,  -11 

smallest  possible  vana- 
tion  in  the  voltage  from 
no  load  to  full  load.  Since  with  non-inductive  load  the  power 
factor  is  unity,  the  power  output  is  found  by  taking  the  product 
of  terminal  voltage  and  external  current. 

§  7.  The  causes  for  the  decrease  in  terminal  voltage  with  load 
are  impedance  drop  in  the  armature  (due  to  its  resistance  and 
inductance)  and  armature  reactions,  discussed  more  fully  in  the 

*  For  other  power  factors,  see  §  13  ;  take  data  as  in  §  14.    Also  see  Fig. 
7,  Exp.  3-B. 


3-A] 


CHARACTERISTICS. 


67 


next  experiment.  Compare  also  §  16,  Exp.  i-B.  As  in  the  case 
of  a  shunt  generator  (§19,  Exp.  I— B),  when  the  iron  is  highly 
saturated,  the  demagnetizing  effect  of  armature  reaction  is 
the  least  and  the  regulation 
the  best.  + 

§8.  The  external  character-  t;||;S> 
istic,  Fig.  4,  is  practically  an  5<3J>£ 
ellipse.*  At  one  end  of  the  - 

RHEOSTAT  _i 

characteristic     (near    Open    cir-      FIG.  3.      Connections  for  loading  an 

cuit),    an    alternator    tends    to  alternator, 

regulate  for  constant  voltage;  at  the  present  day,  this  is  the 
usual  working  part  of  the  characteristic.  At  the  other  end' 
(near  short  circuit),  an  alternator  tends  to  regulate  for  constant 

current.  The  earliest  alter- 
nators were  constructed  for 
such  operation.  Constant 
current  alternators  are  used 
(less  now  than  formerly) 
for  series  arc  lighting.  For 
this  service  an  alternator 
should  have  high  armature 
reaction  so  as  to  limit  the 
current  on  short  circuit  to 
the  desired  value;  a  reac- 
tance external  to  the  arma- 
ture will  serve  equally  well. 


20        40        60        80       100 
ARMATURE  AMPERES 


FIG.  4.  External  characteristic  of  an 
alternator  at  unity  power  factor.  (The 
dotted  parts  of  these  curves  were  cal- 
culated according  to  Exp.  3~B.) 


§9.  Full-load  Saturation 
Curve. — The  machine  is  run 
at  constant  speed  so  as  to  give  its  normal  full-loadf  current  at 
different  field  excitations.  The  connections  are  as  in  Fig.  3.  To 
obtain  the  curve  for  unity  power  factor,  a  non-inductive  resistance 

*  See  discussion  of  Fig.  7,  Exp.  3-6. 

t  (§9a).  Curves  taken  at  intermediate  loads   (one  fourth,  one  half  and 
three    fourths    full    load)    would   lie   between   the   no-load    and    full-load 


68 


SYNCHRONOUS   ALTERNATORS. 


[EXF. 


is  used  as  load;  with  constant  armature  current,  readings  are 
taken  of  terminal  voltage  for  different  field  currents,  and  plotted 
as  in  Fig.  2. 

For  the  first  reading,  adjust  the  field  rheostat  to  its  maximum 
resistance;*  with  field  circuit  open,  reduce  the  load  resistance  to 
zero  (i.  e.,  short-circuit  the  armature  through  the  ammeter)  ; 
close  the  field  circuit  and  adjust  the  field  rheostat  until  the  de- 
sired value  of  armature  current  is  obtained.  For  each  succeed- 
ing reading,  increase  the  load  resistance  by  a  small  step  and  re- 
adjust the  field  rheostat  until  the  desired  value  of  armature 
current  is  again  obtained,  taking  care  that  the  increase  or  de- 
crease in  excitation  is  continuous. 

§  10.  In  Fig.  2,  the  excitation  data  are  as  follows: 


Excitation. 

Volts. 

Amperes. 

Ampere  Turns. 

No  Load. 

Full  Lead. 

6.66 
7-33 

3090 
3401 

575 
627 

525 

575 

A  comparison  of  the  no-load  and  full-load  saturation  curves, 
Fig.  2,  shows  the  following : 

At  constant  excitation,  the  difference  in  the  ordinates  of  the 
two  curves  (their  distance  apart  vertically)  shows  the  difference 
in  terminal  voltage  of  the  alternator  at  no  load  and  at  full  load. 

At  constant  terminal  voltage,  the  difference  in  the  abscissae  of 
the  two  curves  (their  distance  apart  horizontally)  shows  the 
difference  in  excitation  (magnetomotive  force)  required  at  no 
load  and  full  load  in  order  to  maintain  the  voltage  constant. 

At  constant  excitation,  a  voltage  of  575  at  full  load  increases 
to  627  when  the  load  is  thrown  off,  giving  a  regulation  of  9  per 

curves  of  Fig.  2.  To  take  these  is  unnecessary,  unless  some  special  object 
is  in  view.  For  inductive  load,  the  full-load  saturation  curve  will  be  lower 
than  with  non-inductive  load  (as  shown  in  Fig.  I,  Exp.  3-B,  for  zero  power 
factor).  For  different  power  factors,  see  §  13,  and  take  data  as  in  §  14. 

*  This  resistance  should  be  sufficient  to  reduce  the  field  current  to  but  a 
small  fraction  of  its  normal  value. 


3-A] 


CHARACTERISTICS. 


69 


TERMINAL  VOLTAGE,  575 

T3 

3 


4000 


3000 


2000 


1000 


cent.,  the  same  as  already  obtained  from  the  external  character- 
istic, Fig.  4. 

For  constant  terminal  voltage  of  575,  the  excitation  must  be 
increased  from  6.66  amperes  at  no  load  to  7.33  amperes,  at  full 
load.  This  will  be  found  to  check  approximately  with  the  arma- 
ture characteristic,  Fig.  5 ;  an  exact  check  can  not  be  expected. 

Fig.   2   shows   that,   as   we 
go  above  saturation,  there  is      10 
less    difference    between    the 
no-load  and  full-load  voltages,    £ 

UJ 

*.  e.,  the  regulation  is  better    §[ 

(§7). 

§11.  Armature  Character-    c 
istic   or   Field   Compounding 
Curve. — This   curve   is   taken 
for  an  alternator*  in  the  same 
way  as   for  a   direct  current 
generator    (§26,   Exp.    i-B). 
The   curve   in   Fig.    5,   taken 
for  a  constant  terminal  volt- 
age of  575  at  unity  power  factor,  shows  that  in  going  from  no 
load  to  full  load  (43.4  amperes)  the  excitation  is  increased  from 
6.6  to  7.25  amperes.     This  checks  with  the  increase  6.66  to  7.33 
amperes  in  Fig.  2. 

Armature  characteristics  for  lower  power  factors  than  unity 
will  rise  more  rapidly  (§  13). 

*  (§  na).  Composite  Winding. — Although  an  alternator  can  not  be  com- 
pounded by  a  series  winding  carrying  the  line  or  armature  current,  as  in 
the  case  of  a  direct  current  generator  (since  the  field  winding  requires  a 
direct  current  and  the  line  or  armature  current  is  alternating),  the  result 
can  be  accomplished  by  rectifying  part  of  the  alternating  current  and 
passing  it  through  what  is  called  an  auxiliary  field  winding.  Such  an 
alternator  is  said  to  be  composite  wound.  The  alternating  current  to  be 
rectified  is  commonly  derived  from  the  secondary  of  a  transformer, 
through  the  primary  of  which  flows  the  line  or  armature  current;  for 
the  core  of  this  transformer  the  armature  frame  or  spider  is  used.  The 


20          40          60          80 
ARMATURE  AMPERES 

FIG.  5.  Armature  characteristic,  or 
field  compounding  curve ;  unity  power 
factor ;  speed  constant. 


7°  SYNCHRONOUS   ALTERNATORS.  [Exp. 

APPENDIX  I. 
MISCELLANEOUS    NOTES. 

§  12.  Tests  on  Polyphase  Generators. — The  tests  described  above 
may  be  made  on  polyphase  generators  in  the  same  manner  as  on 
single-phase  machines.  The  polyphase  generator  when  loaded  should 
ordinarily  be  given  a  balanced  load,  i.  e.,  one  that  is  divided  equally 
between  the  several  circuits.  Tests  may  also  be  made  by  loading 
down  one  phase  only  and  taking  measurements  on  the  unloaded  as 
well  as  the  loaded  phases. 

In  plotting  curves,  plot  voltage  and  current  per  phase  (the  more 
usual  way)  ;  or,  line  voltage  and  equivalent  single-phase  current.  See 
Exp.  6-A,  particularly  §§  28-30. 

§13.  Power  Factors  Less  than  Unity. — The  characteristics  of  an 
alternator  under  load  vary  with  the  power  factor  of  the  load.  With 
a  power  factor  less  than  unity  and  current  lagging,  the  regulation 
will  be  poorer,  the  full-load  saturation  curve  will  be  lower,  the  exter- 
nal characteristic  lower  and  the  armature  characteristic  higher  than 
at  unity  power  factor.  The  reverse  is  true  when  the  current  is  lead- 
ing (instead  of  lagging),  as  it  may  be  when  there  is  capacity  in  the 
line  or  in  the  load,  or  when  the  load  consists  in  part  of  over-excited 
synchronous  motors  or  converters. 

These  facts  may  be  fully  shown  by  calculation  (Exp.  3~B),  or 
by  a  complete  series  of  runs  made  with  loads  of  different*  power 
factors.  If  such  runs  are  to  be  made,  it  will  be  more  profitable  to 
make  them  after  Exp.  3-6.  At  present,  it  will  suffice  to  illustrate 
these  facts  by  a  few  readings  only,  as  in  the  next  paragraph. 

§  14.  Tests  to  Compare  Effects  of  Inductive  and  Non-inductive 
Loads. — The  difference  between  inductive  and  non-inductive  loads 

composite  winding  is  not,  however,  being  extensively  used,  for  it  can  not 
give  constant  voltage  under  all  conditions — e.  g.,  varying  power  factor — 
and  the  rectifying  commutator  is  liable  to  spark.  The  Tirrell  regulator 
(§3a,  Exp.  i-B),  applied  to  the  exciter  of  an  alternator,  can  maintain 
constant  voltage  under  all  conditions  of  load. 

*  (§  I3a).  This  will  require  special  facilities  for  adjusting  power  factor; 
for  an  inductive  load,  this  can  be  done  by  means  of  an  adjustable  resist- 
ance and  adjustable  reactance  in  parallel.  Runs  should  be  made  at  one 
high  power  factor,  one  medium,  and  one  as  low  as  can  be  obtained. 


3-A]  CHARACTERISTICS.  71 

can  be  illustrated  by  the  following  tests,  or  by  modifications  which 
may  be  devised  by  the  experimenter. 

1.  Load  the  alternator  on  inductive  load,  using  for  this  any  one 
particular  load  which  can  be  conveniently  obtained.     An  induction 
motor  can  be  used  for  a  load,  as  in  commercial  practice;  but  a  choke 
coil  will  serve  fully  as  well. 

With  the  same  speed  and  excitation  as  were  used  in  taking  the 
external  characteristic  on  non-inductive  load,  Fig.  4,  take  readings* 
of  load  current  and  terminal  voltage  with  the  inductive  load.  These 
readings  are  plotted,f  in  Fig.  4,  as  the  point  p,  which  is  one  point  on 
a  characteristic  for  low  power  factor.  (For  more  complete  curves, 
see  Fig.  7,  Exp.  3~B.) 

Throw  off  the  load  and  (  at  the  same  speed  and  excitation)  read 
the  no-load  voltage;  the  per  cent,  increase  in  voltage  when  the  load 
is  thrown  off  gives  the  per  cent,  regulation. 

2.  With  the  same  speed  and  excitation,  repeat  with  a  non-inductive 
load,  so  adjusted  as  to  obtain  the  same  load  current  as  in  i. 

Note  the  terminal  voltage  under  load,  the  no-load  voltage  when  the 
load  is  thrown  off,  calculate  the  regulation  and  compare  with  the 
regulation  in  i. 

3.  With  the   same  speed  and  terminal  voltage  as  were  used   for 
obtaining  the  armature  characteristic  on  non-inductive  load,  Fig.  5, 
note  the   increase   in   field   current   required  with  inductive  load  to 
maintain  constant  terminal  voltage  and  plot  the  point  q,  Fig.  5. 

4.  Repeat  with  a  non-inductive  load   (adjusted  for  the  same  load 
current)  and  compare  results. 

§  15.  Efficiency. — If  the  alternator  is  driven  by  a  direct  current 
motor,  the  friction  and  core  loss  are  conveniently  determined  by  the 
method  of  §  21,  Exp.  2-B.  If  the  driving  motor  is  alternating,  a 
wattmeter  is  used  to  measure  its  input,  the  increase  in  motor  input 

*  (§  143).  If  a  wattmeter  reading  is  also  taken,  the  power  factor  can  be 
found  by  dividing  the  reading  of  the  wattmeter  by  the  product  of  current 
and  voltage. 

t  (§  I4b).  Since  the  same  value  of  exciting  current  may  at  different 
times  give  different  amounts  of  magnetization  (as  in  the  case  of  the 
ascending  and  descending  curves),  the  point  p  thus  located — and  the  point 
q  as  located  later — may  not  be  exact  in  their  positions,  as  compared  with 
the  characteristics  previously  taken.  They  will,  however,  serve  to  illus- 
trate the  effects  in  question. 


7-2  SYNCHRONOUS   ALTERNATORS.  [Exp. 

giving  the  friction  and  core  loss  of  the  alternator — any  changes  in 
motor  losses  being  corrected  for,  if  necessary. 

The  copper  losses  of  field  and  armature  are  calculated  from  resist- 
ance measurements,  and  the  efficiency  so  determined. 

If  the  armature  has  large,  solid  conductors,  the  loss  in  them  will  be 
greater  with  alternating  than  with  direct  current,  this  additional  loss 
being  a  load  loss.  Load  losses  are  losses  which  occur  under  load  in 
addition  to  the  losses  already  accounted  for,  i.  e.,  in  addition  to  core 
loss,  RI2,  friction  and  windage.  There  is  no  simple  and  accurate 
method  for  determining  load  losses  in  alternators.  The  A.  I.  E.  E. 
Standardization  Rules  (116-7)  giye  a  method  for  estimating  these 
losses  by  assuming  them  to  be — in  the  absence  of  more  accurate  infor- 
mation— equal  to  one  third  of  the  short-circuit  core  loss. 


3-B1  PREDETERMINATION.  73 

EXPERIMENT  3-B.  Predetermination  of  Alternator  Charac- 
teristics.* 

§  i.  Introductory. — It  is  desirable  to  be  able  to  predetermine 
the  performance  of  any  machine  without  loading,  and  this  is 
particularly  true  of  alternators;  for,  in  the  case  of  large  ma- 
chines, the  regulation  can  not  be  conveniently  found  in  any 
other  way. 

There  are  two  simple  methods  for  predetermining  the  per- 
formance of  an  alternator  approximately, — the  electromotive 
force  method  and  the  magnetomotive  force  method.  Although 
other  more  complex  methods  are  proposed  for  the  more  exact 
determination,  no  one  method  has  been  found  which  is  generally 
accepted  and  gives  correct  results  in  all  cases.  It  is  well  to  first 
thoroughly  study  the  electromotive  force  method,  on  account  of 
the  insight  it  gives  into  the  general  performance  of  the  alterna- 
tor and  into  other  methods  of  dealing  with  the  subject.  The 
magnetomotive  force  method  should  then  follow;  after  which, 
other  methods  (essentially  modifications  of  these  two)  can  be 
made  a  special  study  by  those  who  desire  to  pursue  the  subject 
further.  (See  Appendices  I.  and  II.) 

§  2.  There  are  primarily  two  causes  for  the  change  in  termi- 
nal voltage  of  an  alternator  with  load : 

1.  The  effect  of  armature  resistance,  which  is  small  and  defi- 
nite ;  this  causes  a  drop  in  electromotive  force  which  is  in  phase 
with  the  armature  current  and  is  equal  to  R  I. 

2.  The  effect  of  the  flux  set  up  by  the  armature  current,  a 
much  larger  and  less  definite  effect,  discussed  in  the  next  para- 
graph. 

§  3.  All  the  flux  set  up  by  the  armature  current  encircles  the 

*To  be  preceded  by  Exp.  4-A.  See  §  9  for  a  statement  of  data  to  be 
taken.  For  a  short  experiment,  take  §§  1-18  and  26-30,  plotting  curves 
for  unity  power  factor  only.  The  curves  used  to  illustrate  this  experi- 
ment and  Exp.  3-A  all  relate  to  the  same  machine. 


74  SYNCHRONOUS   ALTERNATORS.  [Exp. 

armature  conductors.  There  are,  however,  different  paths  which 
the  flux  may  follow,  causing  different  inductive  effects. 

§4.  (a)  True  Armature  Reaction. — By  one  path,  flux  set  up 
by  the  armature  conductors  passes  into  the  pole  pieces  and 
through  the  magnetic  circuit  of  the  field  magnets  (Fig.  10), 
linking  with  the  windings  of  the  field  coils.  This  flux  has  a 
demagnetizing  effect,  weakening*  the  field  by  a  certain  mag- 
netomotive force  produced  by  the  ampere-turns  of  the  armature. 

This  flux  through  the  field  magnets  is  maintained  by  successive 
armature  conductors;  in  a  single-phase  alternator  it  is  pulsating, 
but  in  a  polyphase  alternator,  due  to  the  combined  effect  of  the 
armature  currents  in  the  different  phases,  it  is  constant  both  in 
position  and  in  magnitude. 

§  5.  (b)  Local  Armature  Reactance. — By  a  different  path, 
flux  set  up  by  the  armature  current  encircles  the  armature  con- 
ductors without  entering  the  pole-pieces;  this  flux  (the  fine  lines 
in  Fig.  9)  is  entirely  in  the  armature,  or  partly  in  the  armature 
and  partly  in  the  air  gap.  The  flux  surrounding  any  particular 
conductor  varies  periodically  and  produces  a  reactance  electro- 
motive forc.e  or  reactance  drop,  XI,  in  quadrature  with  the  arma- 
ture current  and  proportional  to  it, — as  in  any  alternating  cur- 
rent circuit. 

§  6.  By  another  and  somewhat  similar  path,  flux  encircles 
the  armature  conductors  by  entering  into  and  returning  from  the 
poles  without  linking  with  the  windings  of  the  field  circuit;  this 
flux  is  shown  by  heavy  lines,  Fig.  9.  This  is  cross-magnetising 
flux  and  distorts  the  field ;  it  does  not  weaken  the  field  except 
incidentally  to  a  small  extent  by  saturating  the  pole  pieces.  This 
cross-magnetization  is  alternating  with  respect  to  the  armature 
conductor,  as  in  (b)  ;  with  respect  to  the  pole  pieces,  it  is  con- 
stant in  a  polyphase  and  pulsating  in  a  single-phase  alternator, 

as  in  (a).    It  may  be  treated  separately;  or  with  (a)  or  (b). 

I 

*  The  field  is  weakened  by  a  lagging  current,  but  strengthened  by  a 
leading  current,  §§  46-8. 


3-B]  PREDETERMINATION.  75 

§7.  It  is  thus  seen  that  there  are  two  somewhat  different 
effects  produced  by  the  armature  current:  the  first  (a)  is  a 
magnetomotive  force,  which  reduces  the  field  flux  and  so  reduces 
the  generated  voltage;  the  second  (b)  is  an  electromotive  force, 
which  is  subtracted  from  the  generated  electromotive  force  (in 
the  proper  phase)  so  as  to  give  a  lower  terminal  voltage. 

These  two  effects  operate  simultaneously  to  lower  the  terminal 
voltage,  the  relative  amounts  of  the  two  varying  according  to 
details  of  design, — saturations,  air-gap,  shape  of  slots,  etc.  To 
take  full  and  accurate  account  of  the  two  effects — treating  one 
as  a  magnetomotive  force  and  the  other  as  an  electromotive 
force — is  difficult*  and  will  not  be  undertaken  here. 

§  8.  We  may,  however,  instead  of  treating  the  two  effects 
separately,  treat  them  combined,  following  either  one  of  two 
methods : 

(a)  The  magnetomotive  force  or  ampere-turn  method,  which 
assumes  that  all  the  effect  is  magnetomotive  force ;  or, 

(b)  The  electromotive  force  or  reactance  method,  which  as- 
sumes that  all  the  effect  is  electromotive  force. 

If  the  saturation  curve  were  a  straight  line,  the  two  methods 
would  be  identical  ;f  for,  magnetomotive  force  would  produce  a 
proportional  electromotive  force.  With  the  saturation  curve, 
however,  not  a  straight  line,  a  given  increase  or  decrease  in  mag- 
netomotive force  will  cause  a  less  than  proportional  change  in 
electromotive  force. 

Hence,  if  we  consider  that  all  the  effect  of  armature  flux  is 
a  magnetomotive  force,  we  will  have  a  less  drop  in  terminal 
voltage  than  if  we  consider  that  all  the  effect  is  an  electromotive 
force.  The  magnetomotive  force  method  is,  accordingly,  opti- 
mistic (Behrend)  and  gives  the  generator  a  better  regulation 

*  See  Appendix  II. 

fThis  would  be  true  if  the  details  of  the  two  methods  were  in  all 
respects  the  same.  Differences  in  the  details  of  the  two  methods,  as 
usually  applied,  cause  differences  in  the  results,  even  though  the  saturation 
curve  is  straight. 


76  SYNCHRONOUS   ALTERNATORS.  [Exp. 

than  it  actually  has ;  the  electromotive  force  method,  on  the 
other  hand,  is  pessimistic,  giving  the  generator  a  poorer  regula- 
tion than  the  actual. 

The  two  methods,  therefore,  give  the  limits  between  which  is 
the  true  performance  of  the  machine. 

§  9.  Data. — For  either  method,  the  data  required  are  obtained 
from  the  following  two*  runs,  which  are  made  without  loading 
the  generator : 

1.  An  open-circuit  run,  giving  the  open-circuit  voltage  EQ, 
for  different  field  currents, — i.  e.,  the  no-load  saturation  curve, 
obtained  as  in  §  5,  Exp.  3~A.     See  Curve  (i),  Fig.  i.     To  save 
labor  in  the  many  subsequent  calculations,   it  is  customary  to 
use  only  the  ascending  curve. 

2.  A  short-circuit  run,  giving  the  short-circuit  current  Is,  for 
different  field   currents, — called   also   a   synchronous   impedance 
test, — as  described  in  the  next  paragraph.    See  Curve  (2),  Fig.  i. 

These  data  enable  us  to  ascertain  the  synchronous  impedance 
of  the  armature  and  hence  to  compute  the  volts  impedance  drop 
for  the  electromotive  force  method ;  they  also  enable  us  to  ascer- 
tain the  magnetomotive  force  required  to  overcome  the  mag- 
netizing effect  of  the  armature,  for  the  magnetomotive  force 
method. 

The  hot  armature  resistancef  is  to  be  found  by  the  fall-of-po- 
tential  method. 

§  10.  Test  for  Short-circuit  Current  and  Synchronous  Impe- 
dance.— With  the  armature  short-circuited  through  an  ammeter,J 

*  (§Qa).  Two  such  runs  are  common  in  testing  many  kinds  of  appa- 
ratus ;  note,  for  example,  the  open-circuit  and  short-circuit  tests  for  trans- 
formers, Exp.  5-B. 

t(§Qb).  On  account  of  eddy  currents,  the  resistance  will  be  greater 
for  alternating  currents  than  the  value  found  by  direct  current.  This  is 
of  importance  as  affecting  efficiency  (§  15,  Exp.  3-A),  but  is  of  little  con- 
sequence so  far  as  regulation  is  concerned,  for  RI  drop  has  only  a  small 
effect  at  high  power  factors  and  is  negligible  at  low  power  factors,  as  will 
be  seen  later. 

£  (§  loa).     The  ammeter  leads  should  be  short  and  heavy;  for,  by  the 


3-B] 


PREDETERMINATION. 


77 


10     11     12     13     14     15    18 

FIELD  AMPERES 


FIG.  i.  No-load  saturation  curve  (i)  and  short-circuit  current  (2  and  3) 
for  different  field  excitations.  Also  full-load  saturation  curves  (4,  5,  and  6) 
for  zero  power  factor,  current  lagging. 

the  short-circuit  current  is  found  for  different  values  of  field 
current.  The  ammeter  should  have  a  range  of  about  three  times 
full-load  current.  The  speed  should  be  normal,  but  special  care 
in  maintaining  constant  speed  is  not  necessary.* 

methods  of  computations  used  later,  any  drop  in  them  is  included  in  the 
impedance  drop  of  the  armature. 

*  (§  lob).  If  facilities  for  varying  the  speed  are  provided,  with  constant 
excitation  vary  the  speed  through  wide  range  and  note  the  practical 
absence  of  change  in  the  short-circuit  current.  Note,  however,  that  the 
open-circuit  voltage  is  proportional  to  speed.  How  are  these  facts  explained? 


78  SYNCHRONOUS   ALTERNATORS.  [Exp. 

Beginning  with  the  field  weakly  excited,  increase  the  field  cur- 
rent by  steps  so  that  the  short-circuit  armature  current  (/s)  is 
increased  from,  say,  J  normal  to  ij  or  2  times*  normal  full-load 
current.  At  each  step  read  field  and  armature  currents  and  plot 
as  in  Curves  2  and  3  of  Fig.  i. 

In  the  short-circuit  test,  we  may  have  either  the  field  or  the 
armature  under  normal  full-load  working  conditions,  but  not 
both  at  the  same  time. 

§  ii.  The  curve  for  short-circuit  current,  will  (as  in  Fig.  i) 
be  a  straight  line  through  a  wide  working  range,  and  may  be 
extended  as  a  straight  linef  beyond  the  observed  data.  The 
ultimate  bending  of  the  curve  depends  upon  the  relative  satura- 
tions of  various  parts  of  the  magnetic  circuit, — armature,  teeth, 
poles,  etc. 

Fig.  i  shows  that  normal  excitation,  OH  =  7.33  amperes, 
gives  a  short-circuit  current  of  116  amperes.  (Normal  excita- 
tion is  the  excitation  giving  rated  voltage,  575,  at  full  load,  unity 
power  factor;  for  this  machine — see  Figs.  6  and  7 — the  corre- 
sponding no-load  voltage  is  found  to  be  627.) 

*  (§  loc).  By  taking  the  run  quickly,  even  higher  values  of  current  can 
be  reached. 

Running  an  alternator  on  short  circuit,  as  described,  affords  the  best 
means  for  drying  armature  insulation.  An  alternator  in  shipment  may 
have  been  unduly  exposed  to  weather  or  have  been  allowed  to  stand  in  a 
damp  place.  The  insulation  readily  takes  up  moisture  and  is  much 
impaired  thereby.  In  such  a  case,  as  soon  as  the  alternator  is  installed 
it  should  be  run  for  one  day  with  the  armature  short-circuited,  the  field 
excitation  being  so  low  that  the  normal  armature  current  flows;  there  is 
no  high  voltage  to  break  down  the  insulation.  The  armature  is  thus  baked 
and  the  insulation  restored.  This  precaution,  particularly  in  the  case  of 
high  voltage  machines,  may  avoid  a  break-down  of  insulation  upon 
starting  up. 

t  (§  na).  Extrapolation  as  a  straight  line  (2)  gives  (after  saturation  is 
reached)  a  diminishing  value  for  synchronous  impedances  Z  =  Eo  -f-  Is, 
as  used  later.  It  thus  favo'rs  the  machine  by  giving  a  smaller  impedance 
drop;  in  the  electromotive  force  method  this  is  justifiable  because  it  par- 
tially offsets  the  pessimistic  tendency  of  that  method.  This  justification 
is  empirical. 

Curve   (3)   has  been  extrapolated  by  assuming  Eo  -r-  Is  to  be  constant. 


3-B] 


PREDETERMINATION. 


79 


An  excitation,  OG  =  2.6  amperes,  is  required  to  cause  normal 
full-load  current  (43.4  amp.)  on  short  circuit.  The  corresponding 
impedance  voltage  is  £2  =  234,  for  on  short  circuit  the  whole 
generated  voltage  is  used  in  overcoming  the  internal  or  armature 
impedance. 

§  12  Synchronous  Impedance. — On  short  circuit,  the  whole 
generated  voltage  is  equal  to  the  internal  impedance  drop  in  the 
armature.  Impedance  is  equal  to  impedance  drop  divided  by 
current;  hence,  the  synchronous  impedance  of  the  armature — 
i.  e.,  its  impedance  when  running  at  synchronous  speed — is  equal 


40  60  80          100          120          140 

AMPERES  ON  SHORT  CIRCUIT:  Ia 

FIG.  2.     Impedance,   reactance,   and  resistance  drop.       (All  the  curves  in 
Exps.  3-A  and  3-6  relate  to  the  same  machine.) 

to  the  generated  voltage  EQ,  divided  by  the  short-circuit  current 
Is. 

For  any  field  current,  the  values  of  EQ  and  Is  are  obtained 
from  curves  (i)  and  (2),  Fig.  i ;  the  corresponding  synchronous 
impedance,  Z  =  Eo-+-Is,  should  be  plotted  as  a  curve  (not 
shown).  It  will  be  found  nearly  constant \ for  a  wide  range, — 
diminishing,  however,  for  high  values  of  field  current. 

§  13.  In  Fig.  2,  the  curve  marked  impedance  drop  is  plotted  by 


. 

So  SYNCHRONOUS   ALTERNATORS.  [Exp. 

taking,  from  Fig.  i,  corresponding  values  for  £0  and  7S. 
Eventually  there  is  a  tendency  for  the  curve  to  bend,  although 
in  this  instance  there  is  none  within  the  range  for  which  Fig.  2 
is  drawn.  The  ratio  of  any  ordinate  to  the  corresponding 
abscissa  gives  the  value  of  the  synchronous  impedance;  thus,  in 
Fig.  2,  the  impedance  drop  is  234  volts  for  a  full-load  current  of 
43.4  amperes,  and  the  impedance  is,  therefore,  234^-43.4  =  5.4 
ohms.  The  normal  full-load  voltage  of  this  machine  is  575 ;  the 
impedance  drop  is,  accordingly,  40.7  per  cent.  This  is  called* 
the  impedance  ratio.  An  open-circuit  voltage  of  627  is  seen  to 
give  a  short-circuit  current  of  116  amperes,  as  already  seen  in 
Fig.  i. 

§  14.  Resistance  drop  is  plotted  as  a  straight  line,  Fig.  2.  The 
resistance,  found  by  the  fall-of -potential  method,  is  0.17  ohms; 
the  resistance  drop,  for  43.4  amperes,  is  0.17  X  434  =  7-4  volts. 

§  15.  The  reactance  drop  is  Ex  —  V-E?- — ER2',  or,  for  43.4 
amperes,  reactance  drop  =  V2342  —  74*  =  233-9  volts.  Usu- 
ally, as  in  this  case,  resistance  is  small  so  that  there  is  little  differ- 
ence between  the  values  of  synchronous  impedance  and  synchro- 
nous reactance.  It  is  common,  therefore,  not  to  calculate  the 
value  of  reactance  drop,  but  to  use  the  value  of  impedance  drop 
in  its  place. 

Synchronous  reactance  is  proportional  to  speed ;  hence,  syn- 
chronous impedance  is  practically  proportional  to  speed. 

Synchronous  impedance  and  synchronous  reactance  are  ficti- 
tious quantities,  comprising  not  only  the  real  impedance  and  re- 
actance of  the  armature,  but  also  including  the  effect  of  arma- 
ture reactions. 

It  is  instructive  to  compare  the  curves  of  Fig.  2  with  similar 
curves  for  a  transformer ;  see  Fig.  7,  Exp.  5— B. 

§  1 6.  Electromotive  Force  Method. — Aside  from  its  usefulness 
in  predetermining  the  performance  of  alternators,  this  method 
serves  as  an  excellent  illustration  of  the  use  of  vector  diagrams 

*  Standardization  Rule,  208. 


3-B]  PREDETERMINATION.  81 

in  solving  alternating  current  problems ;  it  is  a  practical  applica- 
tion* of  the  elementary  principles  discussed  in  detail  in  Exps. 
4-A  and  4-B.  The  electromotive  force  method  is  general,  apply- 
ing to  all  classes  of  alternating  current  problems, — transmission 
lines  (§  56),  transformers 
(Exp.  5-C),  etc.  For  this 
reason  the  method  will  be 
treated  in  considerable  de- 
tail. ^ — •  ft}7 
8  17.  Unity  Pozver  Fac-  ^^  fc »J_ 

O     Z=43.4     '  £T=575  B*C 

tor. — With  a  non-inductive  7 

load,  the  power   factor  of  ^ 

the   load   is   Unity;   the  CUr-  FlG-  3'     E1^rom0tive  force  diagram,  at 

unity  power  factor ;  current  in  phase  with 
rent  which  flows  is,  accord-       terminal  voltage. 

ingly,  in  phase  with  the  ter- 
minal voltage.  This  is  shown  in  Fig.  3,  in  which  the  terminal 
voltage  ET,  is  in  phase  with  the  current  I.  The  armature  resis- 
tance drop,  ER  =  RIf  is  in  the  direction  of — in  phase  with — the 
current  / ;  the  reactance  drop,  Ex  =  XI,  is  in  quadrature  with  7. 
The  total  generated  electromotive  force  EQ,  is  accordingly  the 
vector  sum  of  the  following  three  electromotive  forces:  ET  de- 
livered to  the  load ;  RI  to  overcomef  armature  resistance  and  XI 
to  overcome  armature  reactance. 

*  (§  i6a).  This  application  illustrates  the  way  that  general  principles 
can  be  put  to  practical  purposes ;  the  application  was  first  made  indepen- 
dently, and  more  or  less  simultaneously,  by  ^various  engineers.  The 
writer  used  the  method  in  numerical  problems  to  illustrate  the  elementary 
principles  of  Bedell  and  Crehore's  Alternating  Currents  in  the  early  nine- 
ties soon  after  the  issue  of  that  book,  and  applied  it  a  little  later  to 
laboratory  data.  The  data  and  some  of  the  curves  here  given  are  taken 
from  a  laboratory  outline  prepared  by  the  writer  for  student  use  and 
printed  in  the  Sibley  Journal,  1897-8,  p.  215. 

t(§i7a).  The  arrows  show  the  direction  of  the  vectors  in  the  sense 
that  EC  and  CA  are  electromotive  forces  to  overcome  resistance  and 
reactance,  respectively;  in  the  reverse  sense,  CB  and  AC  are  the  electro- 
motive forces  produced  by  resistance  and  reactance. 

7 


82  SYNCHRONOUS   ALTERNATORS.  [Exp. 

§  18.  Knowing  the  values  of  resistance  drop  RI,  and  reactance 
drop  XI,  we  may  have  either  of  two  problems  to  solve : 

(a)  Given  the  terminal  voltage  ET,  to  determine  the  open- 
circuit  voltage  EQ;  or, 

,i    (b)   Given  the  open-circuit  voltage  EQ,  to  determine  the  termi- 
nal voltage  ET. 

The  following  examples  will  make  clear  the  solution  of  either 
problem. 

(a)  Given  £7  =  575;  #7  =  7.4;  XI  =  233.9.  Required  to 
find  EQ. 

Lay  off  to  scale  the  values  of  ET,  RI  and  XI,  as  in  Fig.  3 ;  by 
construction  EQ  is  found  to  be  627.  Designating  the  total  in- 
phase  voltage  by  Ep,  and  the  quadrature  voltage  by  EQ;  we  have, 
by  computation, 

Eo  = 


=  V  ( 575  +  74)  2  + 233^  =  62?' 

The   regulation   is   9   per   cent.,   EQ   being  9   per   cent,   greater 
than  ET. 

(b)  Given  £0  =  627;  ^7  =  7.4;  XI  —  233.9.  Required  to 
find  ET. 

Lay  off  RI  and  XI  to  scale,  as  in  Fig.  3.  From  A  as  a  center 
and  radius  EQ  =  627,  strike  an  arc  cutting  at  O  the  line  OB, 
drawn  as  a  continuation  of  BC.  By  this  construction,  ET  is 
found  to  be  575 ;  by  computation 

E^  =  V£o2  —  (Xiy  —  RI  =  V6272  — 233.9*  —  74  =  575. 

At  unity  power  factor,  it  is  seen  that  the  terminal  voltage  is 
always  less  than  the  generated  or  no-load  voltage. 

§  19.  Power  Factor  Less  than  Unity,  Current  Lagging. — With 
an  inductive  load,  the  power  factor  of  the  load  is  less  than  unity 
and  the  current,  accordingly,  lags  behind  the  terminal  electro- 
motive force.  This  is  shown  in  Fig.  4  in  which  the  current  / 
lags  behind  the  terminal  electromotive  force  ET  by  an  angle  0  = 
30°,  the  power  factor  of  the  load,  in  this  case,  being  cos  30°  = 
0.866. 


3-B] 


PREDETERMINATION. 


Fig.  4  is  drawn  by  first  constructing  to  scale  the  triangle  ABC, 
with  two  sides  equal  to  RI  and  XI,  respectively,  and  then  laying 


Cos  6  =  1 


Cos  0  -  0.5*  -  „ 


Cos  0-0 

FIG.  4.     Electromotive  force  diagram,  at  power  factor  0.866 ;  current  lagging 
30°  behind  terminal  voltage. 

off  OB  at  an  angle  0  with  BC,  so  that  cos  6  equals  the  power 
factor  of  the  load. 

(a)   Given  Er  =  575,  we  find  by  construction  £0  =  726;  or, 
by  computation 


cos 


RI)*  +  (£T  sin  6  +  XI) 


=  V(575  X  -866  +  7.4)2  +  (575  X  .5  +  233-9)"*  = 

The  regulation  is  26.3  per  cent.  With  inductive  load,  the 
regulation  is  always  poorer  than  with  non-inductive  load.  The 
clotted  quadrant  indicates  the  locus  of  the  point  O  for  different 
power  factors. 

(b)  Given  EQ  and  power  factor;  required  the  terminal  voltage 
ET.  Lay  off  a  line  in  the  direction  BO  making  the  proper  angle  6. 


S4 


SYNCHRONOUS   ALTERNATORS. 


[Exp. 


Strike  an  arc  from  A  as  a  center,  with  a  radius  EQ,  cutting 
the  line  OB  at  O,  thus  giving*  OB  =  ET. 

§  20.  Power  Factor  Less  than  Unity,  Current  in  Advance.  — 
This  case  is  shown  in  Fig.  5.  The  triangle  ABC  is  drawn  as  be- 
fore, and  OB  is  laid  off  making  an  angle  6  with  BC,  so  that  cos  0 
equals  the  power  factor  of  the  load.  The  current  I,  for  this 
case,  is  30°  in  advance  of  ET- 

(a)  Given  £7  =  575,  we  find  by  construction  £0  =  508;  or, 
by  computation, 


cos 


sin  0  —  XI) 


=  V(575  X  .866  +  74)2  +  (575  X  .5  —  233-9)2  =  5°8. 


The  regulation  is 


•12  per  cent. 

Cos£-0_ 


Cos  0-1 


(b)  Given  £0  ;  the 
terminal  voltage  ET  is 
found,  as  before,  by 
striking  an  arc  from  A 
as  a  center,  with  a  ra- 
dius EQ,  cutting  the  line 
OB  at  0. 

For  a  leading  current, 
3    the   terminal   voltage    is 

CN 

^   always  greater  than  for 
a  lagging  current  or  for 
unity  power  factor,  and 
g      may  even  be  equal  to  or 
Electromotive  force  diagram,   at  greater  than  the  no-load 


FIG.    5. 
power   factor   0.866 ;    current   30°    in    advance    voltage. 


of-  terminal  voltage. 


§ 


Zero  Pozver  Factor.  —  At  zero  power  factor,  cos  6  =  0,  sin  6=  i. 

*  (§  ipa).  The  graphical  construction  for  this  case  will  usually  be  pre- 
ferred; an  analytical  expression  for  ET,  derived  from  the  figure,  is 
ET  =  VJSo2—  (XI  COS0  —  RI  sin  0)z  —  (RI  cos  0-\-XI  sin  0). 


3-B]  PREDETERMINATION.  85 

From  Figs.  4  and  5  it  is  seen  that  the  RI  drop  becomes  ineffec- 
tive, being  at  right  angles  to  ET,  and  can  be  neglected.  Hence, 
practically, 

ET  =  EO  —  XI,  for  lagging  current  ; 

ET  =  Eo  +  XI,  for  leading  current. 

For  this  case,  the  various  voltages  are  combined  algebraically. 
Practically,  XI  =  ZI  =  Ez,  and  these  expressions  become 


This  expression,  approximate  for  0  =  90°,  would  be  exact  for 
a  value  of  6  a  little  less  than  90°  ;  so  that,  in  Fig.  4,  OB  A  forms 
a  straight  line  and  tan  0  =  XI-Jr-RI. 

§  22.  Given  the  Terminal  Voltage  at  One  Power  Factor,  to 
Determine  it  at  Any  Other  Power  Factor.  —  Given  ET  at  any 
power  factor,  E0  is  found  by  method  (a)  of  the  preceding  para- 
graphs. With  Eo  thus  known,  the  value  of  ET  is  readily  found 
for  any  desired  power  factor  by  method  (b). 

In  conducting  tests,  it  is  often  difficult  or  impossible  to  deter- 
mine ET  at  unity  or  high  power  factors,  on  account  of  the  power 
required.  The  value  of  ET  can,  however,  be  found  by  test  at  a 
low  power  factor  (§52)  and  then  determined  by  calculation  for 
any  desired  high  power  factor.  Usually  Eo  is  found  by  test  and 
resistance  drop  is  known;  the  reactance  drop  is  not  known.  In 
this  case  the  procedure  is  as  follows  : 

In  Fig.  4,  lay  off  resistance  drop  BC;  at  right  angles  draw  the 
indefinite  line  CA,  —  the  value  of  reactance  drop  being  unknown. 
At  an  angle  B  with  BC,  lay  off  BO  equal  to  the  value  of  ET  found 
by  test  at  power  factor  cos  6.  Draw  OA  =  Eo,  as  found  by  test, 
cutting  CA  at  A..  The  point  A  being  located  and  Eo  known, 
values  of  ET  at  any  power  factor  are  determined  by  method  (&) 
above. 

In  this  manner,  if  the  regulation  is  known  for  one  power  fac- 
tor, it  can  be  calculated  for  any  power  factor.  At  constant 
terminal  voltage,  the  locus  of  the  point  O  will  be  the  arc  of  a 


86  SYNCHRONOUS   ALTERNATORS.  [Exp. 

circle  with  B  as  a  center ;  at  constant  excitation,  EQ  is  constant 
and  the  locus  of  O  is  the  arc  of  a  circle  with  A  as  a  center. 

§  23.  Application  of  Electromotive  Force  Method. — Knowing 
the  armature  resistance  and  synchronous  reactance* — obtained 
from  the  short-circuit  test, — the  electromotive  force  method  can 
be  used  for  predetermining  the  regulation,  the  external  charac- 
teristic and  the  full-load  saturation  curve  for  any  power  factor. 

§  24.  Predetermination  of  Regulation  at  Different  Power 
Factors. — By  method  (a)  of»§§  17-20,  determine  the  open-circuit 
voltage  Eo,  corresponding  to  rated  full-load  voltage  at  rated  full- 
load  current,  for  different  power  factors.  The  values  of  arma- 
ture RI  drop  and  XI  drop  corresponding  to  full-load  current  will 
be  constant  in  all  the  computations,  R  and  X  being  taken  as  con- 
stant.f  Plot  the  values  of  EQ,  thus  obtained,  with  power  factor 
(or  6)  as  abscissae,  as  in  Fig.  6.  This  is  to  be  done  for  lagging 
and  for  leading  currents.  Arrange,  also,  a  scale — as  on  the  right 
of  Fig.  6 — to  show  the  values  of  EQ  as  per  cent,  of  full-load 
voltage. 

§  25.  The  curves  show  the  increase  (or  decrease)  in  voltage 
when  full-load  current  is  thrown  off  at  different  power  factors; 
in  per  cent.,  this  gives  the  regulation.  At  power  factor  i.o,  the 

*(§23a).  Synchronous  reactance  is  practically  equal  to  synchronous 
impedance.  In  Figs.  I  and  2,  synchronous  impedance  is  Z  =  Eo  -f-  Is,  and 
is  more  or  less  constant ;  it  can  be  computed  for  the  value  of  Eo  or  for 
the  value  of  Is  corresponding  to  working  conditions. 

Thus,  for  normal  field  excitation,  corresponding  to  Eo  =  627,  we  obtain 
Z  — 627-4- 116  =  5.4  ohms;  the  armature  current  116  amp.  is,  however, 
far  above  normal. 

For  normal  full-load  current,  43.4  amp.,  we  obtain  Z  =  234  -f-  43.4  =  5.4 
ohms;  in  this  case  the  field  excitation  is  far  below  normal. 

It  is  thus  seen  that  Z  can  be  computed  from  the  short-circuit  test  either 
for  normal  field  current  or  for  normal  armature  current;  but  field  and 
armature  currents  can  not  simultaneously  be  normal.  When  Z  is  constant, 
the  two  computations  give  identical  results.  When  Z  is  not  constant,  the 
two  computations  give  different  results ;  either  may  be  used,  but  it  is 
justifiable  to  use  the  method  which  gives  the  smallest  value  for  Z  as 
being  least  pessimistic.  (See  §§  na  and  33.) 

t  See  §  26a. 


3-B] 


PREDETERMINATION. 


87 


regulation  is  9  per  cent.;  at  power  factor  0.5  (lagging  current), 
it  is  37  per  cent. ;  at  power  factor  o.o,  it  is  40  per  cent.  At  high 
power  factors,  it  is  seen  that  a  small  change  in  power  factor 
causes  a  marked  change  in  regulation ;  while  at  lower  power  fac- 
tors the  regulation  is  nearly  constant.  The  reason  for  this  will 
appear  from  a  consideration  of  the  construction  in  Figs.  4  and  5. 
This  fact  is  made  use  of  in  §  52. 


joad  voltage,  (lagging  current) 


1.0       0.9      0.8      0.7     O.C      0.5      0.4      0.3     0.2      0.1 

POWER.  FACTOR 

FIG.  6.     Curves  showing  no-load  voltage  corresponding  to  a  constant  full-load 
voltage  (575)  for  full-load  current  (43.4  amperes)  at  different  power  factors. 

§  26.  Predetermination  of  External  Characteristics. — For  a 
definite  open-circuit  voltage  EQ  and  various  power  factors,  com- 
pute (by  method  (b)  of  §§  17-20)  the  terminal  voltage  ET,  for 
different  load  currents.  Armature  XI  drop  and  RI  drop  are  to 
be  taken  as  proportional  to  current;  i.  e.,  X  and  R  are  taken  as 
constant.*  Data  are  thus  obtained  for  plotting  the  complete  ex- 
ternal characteristic,  from  open  circuit  to  short  circuit,  for  differ- 
ent power  factors. 

*  (§26a).  In  §§  24,  26  and  32,  the  same  constant  values  of  X  and  Z  are 
to  be  used.  In  §  26  it  is  proper  that  X  and  Z  be  considered  constant  for 
the  reason  that  field  excitation  is  constant.  In  §  24  the  armature  current 
is  constant,  but  not  the  field,  and  strictly  speaking  X  and  Z  might  not 
remain  constant,  although  for  simplicity  and  for  ease  in  comparison  they 
are  so  taken.  In  the  case  of  §  32,  X  and  Z  should  only  be  taken  as  con- 
stant for  a  certain  range,  and  for  very  high  saturations  should  be  taken  as 
variable  as  in  §  33. 


88 


SYNCHRONOUS   ALTERNATORS. 


[Exp. 


§  27-  Fig.  4,  Exp.  3-A,  shows  the  characteristic  for  unity 
power  factor.  Power  is  zero  on  open  circuit  and  on  short  cir- 
cuit. Maximum  power  is,  in  this  case,  obtained  at  about  twice 
full-load  current ;  at  short  circuit,  the  current  is  about  2  J  times 
full-load  current.  A  small  short-circuit  current*  is  an  element 

'  1200P 


0          10       20       30       40       50       60       70       80       90      100     110      120     130 

AMPERES  ARMATURE 
FIG.   7.     External   characteristics   at   different   power   factors. 

of  safety,  obtained,  however,  by  large  impedance  drop  and  poor 
regulation.      Compare  §  24%,  Exp.  5~C. 

§  28.  External  characteristics  for  different  power  factors,  with 
current  lagging  and  leading,  should  be  plotted  as  in  Fig.  7.  The 
lowest  possible  characteristic  is  a  straight  line;  it  is  obtained  for 
a  power  factor  (cos  0)  of  such  a  value  that  tan  6=  (armature 
reactance-drop)  -r-  (armature  resistance-drop).  See  §21.  The 

*(§27a).  This  is  the  working  part  of  the  characteristic  for  constant 
current  operation,  see  §  8,  Exp.  3-A.  The  armature  should  have  a  high 
reactance  for  constant  current  and  low  reactance  for  constant  potential. 


3-B1  PREDETERMINATION.  89 

characteristic  for  zero  power  .factor  is  a  little  higher  than  the 
straight  line  for  the  limiting  case ;  the  difference,  however,  is  in- 
appreciable. 

When  the  scale  used  is  such  that  the  ordinate  on  open  circuit 
is  equal  to  the  abscissa  on  short  circuit,  the  characteristics  are 
ellipses  with  a  45°  line  as  axis  (Steinmetz,  Alternating  Current 
Phenomena,  3d  ed.,  p.  304). 

In  any  alternator,  armature  resistance  is  small  and  armature 
reactance  relatively  large,  so  that  the  armature  impedance  is 
practically  all  reactance;  this  gives  curves  as  in  Fig.  7.  If  the 
conditions  were  reversed,  resistance  being  large  and  reactance 
negligible,  the  curves  for  cos  6  =  i  and  cos  0  =  o  would  have  to 
be  interchanged.  Unity  power  factor  would  give  the  poorest 
regulation  and  the  straight  line  characteristic  now  obtained  for 
zero  power  factor ;  for,  with  reactance  zero,  ET  =  £O  —  RI,  in 
place  of  ET  =  EQ  —  XI,  as  in  §21. 

§  29.  Predetermination  of  Full-load  Saturation  Curve  from 
No-load  Saturation  Curve. — By  method  (b)  of  §§  17—20,  com- 
pute the  terminal  voltage  E?  corresponding  to  the  different  open- 
circuit  voltages  of  the  no-load  saturation  curve ;  this  is  to  be 
done*  for  full-load  current  at  unity  power  factor  and  at  zero 
power  factor,  current  lagging.  In  this  manner,  full-load  satura- 
tion curves  are  plotted  for  unity  power  factor  (Fig.  2,  Exp.  3- A) 
and  for  zero  power  factor  (Fig.  I  of  this  experiment). 

§  30.  The  interpretation  of  the  full-load  saturation  curve  for 
unity  power  factor  is  given  in  §§  10,  Exp.  3~A.  The  curve  for 
zero  power  factor  is  capable  of  similar  interpretation.  It  is  seen 
that,  for  the  same  terminal  voltage,  the  excitation  must  be  much 
greater  at  zero  than  at  unity  power  factor;  or,  for  the  same 
excitation,  the  terminal  voltage  is  much  lower. 

§  31.  In  determining  the  full-load  saturation  curves  for  any 
power  factor,  X  and  Z  can  be  taken  as  they  are  (somewhat  vari- 


*  It  is  unnecessary  to  construct  intermediate  curves  for  part  load  and 
for  other  power  factors,  unless  a  special  study  is  to  be  made. 


90  SYNCHRONOUS   ALTERNATORS.  [Exp. 

able,  §33)  or  they  can  be  assumed  constant,*  §32.  The  com- 
putations can  be  readily  made  by  either  method ;  it  is  only  above 
saturation  that  the  results  differ.  This  will  be  discussed  in 
greater  detail  in  the  case  of  zero  power  factor. 

§32.  For  zero  power  factor,  the  terminal  voltage  (§21)  is 
ET  =  EQ  —  Ez ;  that  is,  the  impedance  drop,  Ez  is  subtracted 
arithmetically  from  E. 

In  Fig.  i,  if  impedance  drop  Ez  is  taken  as  constant,  we  obtain 
Curve  (4)  differing  from  Curve  (i)  by  a  constant  distance  (Ez) 
vertically.f  This  is  satisfactory  below  saturation,  but  above 
saturation  is  too  pessimistic. 

§  33.  If  we  wish  to  extend  the  curve  above  saturation,  it  is 
better  to  take  a  variable  value,  Z  =  EQ  -f-  7s,  computed  from 
Curves  (i)  and  (2),  Fig.  i,  for  each  value  of  £0, — that  is,  for 
each  excitation.  This  gives  a  decreasing  value  for  Z  and  results 
in  Curve  (5)  instead  of  (4).  Instead  of  subtracting  from  Curve 
( i )  a  constant  Ez,  we  now  subtract 

Ez  =  ZI  =  ~-Eo. 
Is, 

Here  7  is  full-load  current  (43.4  amp.)  ;  EQ  is  taken  from 
Curve  ( i )  and  7s  is  the  corresponding  short-circuit  current  from 
Curve  (2).  The  formula  can  be  interpreted  thus:  if  a  current 
7s  uses  up  in  the  armature  a  voltage  EQ,  a  current  7  will  use  up 
a  proportional  voltage,  Ez=  (I-~Is)Eo> 

*  See  §  26a. 

fBy  the  magnetomotive  force  method  (Appendix  I.),  Curve  (6)  differs 
from  Curve  (i)  by  a  constant  distance  (A/z)  horizontally;  at  high  satu- 
rations this  is  too  optimistic. 


3-B]  PREDETERMINATION.  91 

APPENDIX  I. 
MAGNETOMOTIVE    FORCE    METHOD.* 

§  34.  In  the  magnetomotive  force  method,  instead  of  combining 
vectorially  various  electromotive  forces — as  was  done  in  the  electro- 
motive force  method,  Figs.  3,  4  and  5 — the  corresponding  magneto- 
motive forces  are  so  combined. 

§  35.  The  magnetomotive  force  corresponding  to  any  electromotive 
force  is  found  by  reference  to  the  no-load  saturation  curve,  and  is 
commonly  expressed  in  ampere-turns.  For  a  given  machine,  with 
constant  number  of  field  turns,  field  amperes  are  proportional  to 
field  ampere-turns  and  may  be  used  as  a  measure  of  magnetomotive 
force.  In  Fig.  I  of  this  experiment  and  Fig.  2  of  Exp.  3~A,  it  is 
seen,  for  example,  that  627  .volts  corresponds  to  a  field  excitation  of 
7.33  field  amperes,  or  3,401  field  ampere-turns,  either  of  which  may 
be  taken  as  a  numerical  measure  of  magnetomotive  force. 

§  36.  It  is  readily  seen  that  a  straight  saturation  curve  gives  mag- 
netomotive forces  proportional  to  electromotive  forces,  so  that  the 
same  results  will  be  obtained  from  the  use  of  either,  if  the  procedure 
is  otherwise  identical.  On  the  other  hand,  a  saturation  curve  which 
is  not  straight  gives  values  of  magnetomotive  forces  not  proportional 
to  electromotive  forces,  so  that  different  results  will  be  obtained 
according  to  whether  magnetomotive  forces  or  electromotive  forces 
are  used. 

§  37.  Method.f — The  three  magnetomotive  forces  Mo,  Mz  and  MT 
are  combined  vectorially,  as  in  Fig.  8;  cos  0  is  the  power  factor  of 
the  load. 

These  three  quantities  Mo,  Mz  and  MT  may  be  interpreted  by  their 
correspondence:}:  to  the  three  electromotive  'forces  Eo,  Ez  and  ET, 

*  No  additional  data  are  required ;  see  §  43  for  the  particular  application 
of  the  method  to  be  made. 

t  (§37a).  This  is  the  common  interpretation  of  the  method  (see  Rush- 
more,  p.  740,  Vol.  I.,  St.  Louis  Elect  Congress,  1904).  In  Franklin  & 
Esty's  Electrical  Engineering,  Mo  is  obtained  as  the  resultant  of  two  mag- 
netomotive forces  which  correspond  not  to  ET  and  Ez,  but  to  Ep  and  EQ 
(the  in-phase  and  quadrature  components  of  £o). 

$(§37b).  If  the  saturation  curve  were  a  straight  line  and  magneto- 
motive forces  were  proportional  to  electromotive  forces,  the  triangles  for 
magnetomotive  forces  and  electromotive  forces  would  be  similar  and  each 
side  of  one  triangle  would  be  perpendicular  to  the  corresponding  side 
of  the  other. 


SYNCHRONOUS  ALTERNATORS. 


[Exp. 


respectively.  A  magnetomotive  force  MT  is  required  for  a  terminal 
voltage  ET,  corresponding  values  being  taken  from  the  saturation 
curve;  at  no  load  no  other  magnetomotive  force  is  required.  Under 
load,  an  additional  magnetomotive  force  BA  =  Mz  is  required  to 
overcome  the  magnetizing  effect  of  the  armature.  In  terms  of  mag- 
netomotive force,  Mz  is  equal  to  the  ampere-turns  of  the  armature; 


p 

Cos  6-0 
FIG.  8.     Magnetomotive  force  method. 

in  terms  of  its  corresponding  electromotive  force,  it  is  a  magneto- 
motive force  which  will  produce  an  electromotive  force  equal  to  the 
armature  impedance  drop,  Ez.  The  total  magnetomotive  force  which 
the  field  must  provide  is  the  vector  sum,  MQ.  In  this  sense,  Mo  is  the 
resultant  of  MT  and  Mz  (  —BA},  in  the  same  way  that  Eo  is  the 
resultant  of  ET  and  Ez. 

Interpreting  these  quantities  further  as  magnetomotive  forces: 
Mo  is  the  magnetomotive  force  produced  by  the  field;  Mz  (=AB,  in 
the  direction  of  armature  current,  /)  is  the  magnetomotive  force 
produced  by  the  armature ;  MT  is  the  combined  magnetomotive  force 
and  produces  the  electromotive  force  ET.  In  this  sense,  MT  is  the 
resultant  of  Mo  and  Mz  (  =AB).  On  open  circuit  the  field  ampere- 


3-B]  PREDETERMINATION.  93 

turns  (or  amperes)  give  us  the  value  of  the  magnetomotive  force 
MT;  for,  in  this- case,  Mz  =  O. 

On  short  circuit,  the  field  ampere-turns  (or  amperes)  give  us  the 
value  of  Mz;  for,  in  this  case,  MT  =  O.  That  is,  on  short  circuit 
the  field  and  armature  ampere-turns  are  (practically)  equal  and  oppo- 
site (compare  §  21). 

In  Fig.  i  it  is  seen  that,  on  short  circuit,  full-load  current  (43.4 
amp.)  is  given  by  a  magnetomotive  force  Mz  =  OG  =  121  ampere- 
turns  (2.6  amperes)  ;  the  corresponding  impedance  voltage,  as  used 
in  the  electromotive  force  method,  is  Ez  =  GF  =  234. 

§  38.  Procedure;  Any  Power  Factor. — The  value  of  Mz  is  known, 
as  in  the  preceding  paragraph ;  also  the  power  factor,  cos  0,  of  the 
load. 

Given  ET  to  find  EQ.  Construct  the  triangle  OBA,  Fig.  8,  from 
the  known  values  of  Mz  and  cos  0,  and  the  value  of  MT  corresponding 
to  ET;  the  value  of  Mo  and  the  corresponding  value  of  EQ  is  thus 
determined. 

Given  E.Q,  the  converse  procedure  is  followed  to  obtain  ET. 

The  most  important  cases  are  for  unity  and. zero  power  factors. 

§  39.  Unity  Power  Factor. — For  this  case,  cos  0=  i,  and  OBA  (Fig. 
8)  becomes  a  right  triangle.  The  same  procedure  is  followed  as  in 
the  preceding  paragraph. 

§  40.  The  following  procedure,  known  as  the  Institute*  Method 
(proposed  by  a  committee  but  not  adopted)  differs  from  the  fore- 
going by  taking  special  account  of  the  armature  RI  drop.  Armature 
RI  drop  is  significant  at  unity  power  factor;  it  becomes  less  so  as 
the  power  factor  decreases  and  becomes  negligible  at  zero  power 
factor.  The  Institute  Rule  is  : 

"  When  in  synchronous  machines  the  regulatio'n  is  computed  from  the 
terminal  voltage  and  impedance  voltage,  the  exciting  ampere-turns  corre- 
sponding to  terminal  voltage  plus  armature  resistance-drop,  and  the 
ampere-turns  at  short-circuit  corresponding  to  the  armature  impedance- 
drop,  should  be  combined  vectorially  to  obtain  the  resultant  ampere-turns, 
and  the  corresponding  internal  e.m.f.  should  be  taken  from  the  saturation 
curve." 

By  the  reverse  procedure  ET  is  determined  when  EQ  is  known. 
*Rule  71,  p.  1087,  Vol.  XIX. 


94  SYNCHRONOUS   ALTERNATORS.  [Exp. 

§41.  Zero  Power  Factor. — When  cos#  =  o,  it  is  seen  that,  by  the 
construction  of  Fig.  8,  Mz  and  MT  are  in  one  straight  line;  hence 

MT  =  Mo  —  Mz ;   or,  Mo  =  MT  +  Mz. 

At  no  load  MO  =  MT.  Under  load,  if  Mr  (and  ET}  is  to  have  the 
same  value  as  at  no  load,  the  field  excitation  Mo  is  to  be  increased  by 
an  amount  Mz  added  in  this  case  arithmetically* 

§  42.  Determination  of  Full-load  Saturation  Curve. — Given  the  no- 
load  saturation  curve,  Fig.  I ;  the  full-load  saturation  curve  for  zero 
power  factor  is  found  by  adding  the  constant  magnetomotive  force 
Mz  =  OG.  The  two  curves  (i)  and  (6)  are  accordingly  a  constant 
distance  apart,  measured  horizontally. 

§  43.  Application. — To  illustrate  the  use  of  the  magnetomotive 
force  method,  it  will  suffice  to  apply  the  method,  using  observed  data, 
to  the  following  typical  cases : 

1.  Using  the  Institute  Method,  §  40,  obtain  EQ,  corresponding  to 
rated  voltage,  ET,  at  full  load,  unity  power  factor.     Plot  this  as  the 
point  p,  Fig.  6.     Note  that  this  point  is  a  little  lower  than  Eo  obtained 
by  the  electromotive  force  method,  i.  e.,  the  regulation  is  better. 

2.  Also,  locate  p  by  the  method  of  §  39. 

3.  By  the  method  of  §§  38  and  41,  locate  the  point  q,  Fig.  6,  that  is 
Eo  corresponding  to  rated  ET  at  full  load,  zero  power  factor.     Note 
that  this  is  considerably  lower  than  Eo  obtained  by  the  electromotive 
force  method. 

4.  Construct  a  full-load  saturation   curve    (§42)    for  zero  power 
factor. 

§  44.  Justification  of  the  Magnetomotive  Force  Method. — The  con- 
struction of  Fig.  8  shows  that  the  armature  ampere-turns  are  com- 
bined with  the  field  ampere-turns  in  such  a  way  as  to  have  the  great- 
est effect  for  power  factor  zero,  cos#  =  o;  the  least  effect  for  cos 
0=  i ;  and  intermediate  effects  for  intermediate  values  of  cos  9.  This 
will  be  shown  to  be  qualitatively  correct,  although  quantitatively  it  is 
only  correct  approximately  or  under  certain  assumptions. 

§  45.  Fig.  9  shows  two  conductors  of  an  armature  coil,  one  midway 
under  a  north  pole,  the  other  midway  under  a  south  pole.  In  this 
position  the  electromotive  force  induced  in  the  armature  conductors 

*  (§4ia).  The  corresponding  electromotive  forces  at  zero  power  factor 
are  likewise  added  arithmetically;  Eo  =  £T  -f-  Ez.  (See  §  21.) 


3-B] 


PREDETERMINATION. 


95 


is  a  maximum.  The  armature  current  will  likewise  be  a  maximum, 
if  it  is  in  phase  with  this  electromotive  force.  In  this  position,  the 
flux  set  up  by  the  armature  current  has  a  cross-magnetizing  effect; 
the  flux  passes  transversely  through  the  pole  piece  but  does  not  pass 
through  or  link  with  the  field  winding  and  so  does  not  directly  oppose 
the  field  ampere-turns. 

Fig.  10  shows  the  armature  conductors  midway  between  poles;  the 
coil,  to  which  these  conductors  may  be  assumed  to  belong,  is  exactly 
opposite  a  pole.  In  this  position  the  electromotive  force  induced  in 


FIG.  9.  Distortion  of  field  by 
transverse  magnetization,  or  cross- 
magnetizing  effect  of  armature  cur- 
rent ;  produced  by  an  in-phase  cur- 
rent, or  component  of  current. 


FIG.  10.  Weakening  of  field  by  de- 
magnetizing effect  of  armature  current ; 
produced  by  a  wattless  or  quadrature 
current,  or  component  of  current. 


the  armature  conductors  is  zero;  at  zero  power  factor  the  armature 
current — lagging  90°  behind  the  electromotive  force — is  a  maximum. 
It  will  be  seen  from  the  figure  that  in  this  position  the  armature  has 
the  greatest  demagnetizing  effect,  the  flux  produced  by  the  armature 
passing  through  the  field  winding  and  directly  opposing  the  field 
ampere-turns. 

§  46.  It  is  seen  that  when  the  armature  current  is  in  phase  with  the 
generated  electromotive  force  it  produces  distortion  and  cross-mag- 
netization ;  when  the  armature  current  is  in  'quadrature  it  produces 
demagnetization  without  distortion,  the  armature  ampere-turns  being 
in  direct  opposition  to  the  field  ampere-turns. 

When  the  current  has  a  phase  displacement,  with  respect  to  the 
induced  electromotive  force,  between  o°  and  90°,  it  may  be  considered 
as  composed  of  two  components,  an  in-phase  component  producing 
cross-magnetization  and  a  quadrature  component  producing  demag- 
netization. 

§  47.  On  short  circuit,  the  current  in  the  armature  lags  90°    (or 


96  SYNCHRONOUS   ALTERNATORS.  [Exp. 

nearly  so,  on  account  of  high  armature  reactance).  The  armature 
and  field  ampere-turns  on  short  circuit  are,  therefore,  practically  equal 
and  opposite.  If  they  were  exactly  equal  and  opposite,  there  would 
be  no  electromotive  force  generated ;  as  a  matter  of  fact,  there  is  a 
very  small  electromotive  force  equal  to  the  armature  RI  drop. 

That  the  armature  ampere-turns  due  to  a  current  lagging  90° 
opposes  or  weakens  (and  does  not  aid  or  strengthen)  the  field  is 
verified  by  this  short-circuit  test,  and  its  resultant  small  electromotive 
force. 

§  48.  A  leading  current,  on  the  other  hand,  directly  aids  and 
strengthens  the  field. 

§  49.  In  the  foregoing  discussion  of  Figs.  9  and  10,  the  reaction 
of  the  armature  has  been  considered  for  the  particular  moment  and 
position  when  the  armature  current  is  a  maximum.  In  reality,  the 
armature  assumes  successively  all  positions  and  the  current  takes  all 
values;  in  intermediate  positions,  demagnetization  and  cross-magneti- 
zation are  both  present  in  varying  amounts  dependent  upon  the  posi- 
tion of  the  armature  and  the  armature  current  at  any  instant.  The 
general  nature  of  the  reaction,  however,  may  be  considered  as  defined 
by  its  character  when  the  current  is  a  maximum.  The  real  effect  is 
a  summation  of  the  effects  at  each  instant  through  a  cycle.  A  more 
complete  discussion  would  involve  some  knowledge  or  assumption  as 
to  flux  distribution  in  the  pole  pieces,  and  other  design  factors. 

As  a  matter  of  fact,  a  sinusoidal  flux  distribution  has  been  assumed 
in  order  to  make  it  possible  to  treat  Mo  as  a  vector  in  Fig.  8;  the 
assumption  tacitly  made  is  that  the  field  flux  passing  through  an  arma- 
ture coil  varies  as  a  sine  function  of  time,  so  that  the  generated  elec- 
tromotive force  (<?  =  —  dQ-s-dt)  is  also  a  sine  function  differing  in 
phase  by  90°.  This  assumption  justifies  the  treatment  of  Mo  and  Eo 
as  vectors  at  90°. 

But  distortion,  by  its  very  nature,  disturbs  the  flux  distribution  and 
makes  the  assumption  necessarily  an  impossible  one.  No  diagram 
using  plane  vectors  can  exactly  represent  all  the  quantities.  The 
justification  of  the  magnetomotive  force  method  is,  therefore,  partly 
empirical.  It  is  found  to  give  fairly  good  result  on  many  modern 
alternators  in  which  armature  reaction  is  large  as  compared  with 
armature  reactance  and  in  which  too  high  saturation  is  not  reached; 
it  is  least  accurate  in  alternators  with  high  saturation  and  relatively 
large  armature  reactance. 


3-B1  PREDETERMINATION.  97 

APPENDIX  II. 
OTHER    METHODS. 

§  50.  There  are  a  number  of  methods  for  determining  the  regula- 
tion and  characteristics  of  alternators  which  are  essentially  modifi- 
cations of  the  electromotive  force  and  magnetomotive  force  methods, 
or  a  combination  of  the  two;  these  methods  are  based  on  test  data 
alone  (obtained  from  open-circuit  and  short-circuit  tests,  §9),  on 
design  data  alone,  or  partly  on  design  and  partly  on  test  data. 
Methods  based  on  design  data  are  of  particular  interest  to  the  design- 
ing engineer  but  cannot  be  taken  up  here;  they  include  methods  for 
calculating  armature  reaction  and  reactance  and  for  predetermining 
the  behavior  of  a  machine  before  its  construction.  (For  further  dis- 
cussion, see  references,  §  55.) 

In  all  methods  use  is  made  of  the  fundamental  principles  brought 
out  in  the  electromotive  force  and  magnetomotive  force  methods, 
which  should  therefore  be  carefully  studied  before  other  methods  are 
undertaken.  For  those  whose  object  is  a  general  understanding  of 
the  behavior  of  alternators,  a  study  of  these  two  methods  is  sufficient; 
but  those  who  desire  to  pursue  the  subject  further  should  consult 
the  references  in  §  55.  It  has  been  pointed  out  that,  so  far  as  results 
are  concerned,  these  two  methods  give  the  pessimistic  and  optimistic 
limits.  Other  methods  give  intermediate,  and  in  some  cases  more 
correct  results;  there  is,  however,  no  one  absolutely  correct  method. 
In  reference  to  this,  Mr.  Behrend  says : 

"  It  appears  wise  to  admit  the  existing  dilemma.  The  question  of  accu- 
rately determining  the  regulation  of  alternators  can  not  be  solved.  .  .  . 
It  seems  to  the  speaker  far  more  dignified  and  more  in  accordance  with 
the  science  that  we  are  working  in,  to  say  that  this  case  is  so  complex,  so 
intricate,  there  are  so  many  factors  to  be  taken  into  account,  that  it  can  no 
more  be  solved  than  you  can  sfate  to  one  thousandth  of  an  inch  the  dis- 
tance between  two  chalk  marks  drawn  on  the  floor."  (A.  I.  E.  E.,  Vol. 
XXIII.,  p.  326.) 

§  51.  Test  Methods. — The  aim  in  various  methods  is  to  test  the 
alternator  under  real  or  equivalent  load  conditions  with  only  a  small 
expenditure  of  power.  The  machine  may  be  actually  loaded  and  the 
power  returned  by  some  opposition  method  (§§27,  273,  Exp.  2-B), 
or  it  may  be  tested  without  any  load  by  simulating  working  load 


98  SYNCHRONOUS   ALTERNATORS.  [Exp. 

conditions.  In  the  preceding  pages  this  was  done  by  two  tests,  the 
open-circuit  test  at  normal  voltage  and  zero  current,  and  the  short- 
circuit  test  at  normal  current  and  zero  voltage,  in  each  test  the  power 
output  being  zero.  But,  inasmuch  as  power  output  is  the  product  of 
current,  voltage  and  power  factor,  E  and  7  may  simultaneously  have 
normal  full-load  values  without  involving  expenditure  of  power  if 
the  power  factor  is  zero.  This  leads  to  the  low  power  factor  tests 
(§  52)  and  split  field  tests  (§  53),  concerning  which  only  a  brief  state- 
ment will  be  made;  for  fuller  information  consult  references.  These 
tests  are  used  in  heat  runs  and  efficiency  tests,  as  well  as  in  test  for 
the  determination  of  regulation. 

§  52.  Tests  at  Low  Power  Factor. — When  operated  at  low  power 
factor,  an  alternator  may  have  full-load  current  and  normal  voltage 
with  only  a  small  expenditure  of  energy.  If  EQ  and  ET  are  thus 
determined  for  one  power  factor,  their  values  and  the  regulation  can 
be  calculated  (§22)  for  unity  or  any  other  power  factor.  This  cal- 
culation is  usually  made  either  for  the  same  terminal  voltage  or  for 
the  same  excitation  (same  £o).  The  load  may  consist  of  react- 
ances, unloaded  induction  motors  or  a  synchronous  motor  with  low 
or  no-field  excitation.  The  power  factor  is  known  from  readings  of 
ammeter,  voltmeter  and  wattmeter.  Any  power  factor  less  than  0.20 
or  0.25  may  be  considered  as  zero,  for  between  these  limits  (see  Fig.  6) 
there  is  practically  no  change  in  regulation. 

When  a  synchronous  motor  is  used,  the  generator  voltage  is  adjusted 
by  the  field  rheostat  of  the  generator;  the  armature  current  by  the 
field  rheostat  of  the  motor.  In  this  way  a  full-load  saturation  curve 
for  low  power  factor  can  be  obtained  (Fig.  i)  and  compared  with 
the  no-load  curve ;  or  points  can  be  plotted  for  an  external  charac- 
teristic, as  in  Fig.  7. 

§  53.  Split  Field  Method. — When  an  alternator  is  operated  at  low 
power  factor  with  a  synchronous  motor  load,  as  in  the  preceding 
paragraph,  electric  energy  is  given  out  by  the  alternator  to  the  motor 
one  quarter-cycle  and  is  practically  all  returned  the  next  quarter- 
cycle;  power  circulates  between  the  two  machines.  Circulation  of 
power  in  one  machine  was  first  proposed  by  Mordey*;  this  was 
accomplished  by  dividing  the  armature  coils  in  two  parts,  one  opposed 
to  the  other.  In  this  way  part  of  the  armature  acted  as  a  generator 
and  part  as  a  motor.  This,  however,  proved  open  to  objection. 

*  W.  M.  Mordey,  Journal  Brit.  Inst.  of  Elect.  Eng'rs,  Vol.  II.,  1893. 


3-B]  PREDETERMINATION.  99 

Behrend  (see  his  St.  Louis  paper,  §  55)  has  developed  a  method 
for  circulating  power  in  one  machine  by  dividing  not  the  armature 
but  the  field  and  reversing  the  excitation  of  one  part  of  the  field. 
The  armature  acts  as  a  generator  with  respect  to  one  part  of  the  field 
and  as  a  synchronous  motor  with  respect  to  the  other  part.  Each 
part  of  the  field  has  its  own  rheostat,  one  controlling  the  generator 
and  the  other  the  motor  action.  Tests  are  made  in  much  the  same 
way  as  though  two  machines  were  used,  §  52.  For  a  later  modifica- 
tion of  this  method,  see  paper  by  S.  P.  Smith,  §  55. 

§  54.  Arguments  for  and  Against  Specifying  Regulation  at  Zero 
Power  Factor. — The  opinion  is  growing  among  engineers  that  regula- 
tion should  be  specified  at  zero  power  factor.  Tests  at  unity  power 
factor  are  objectionable,  not  only  on  account  of  the  use  of  much 
power  which  may  be  prohibitive,  but  also  on  account  of  errors  in  the 
results.  In  Fig.  7  it  is  seen  that  the  difference  in  regulation  for  a 
small  change  in  power  factor  is  very  small  near  zero  power  factor, 
but  is  considerable  near  unity  power  factor. 

At  unity  power  factor,  therefore,  any  inductance  or  capacity  in  the 
load  introduces  a  large  error.  The  use  of  a  water  rheostat  as  a  load 
causes  an  error  for  this  reason,  for  it  possesses  a  capacity  which, 
though  small,  is  sufficient  to  give  an  alternator  a  better  regulation 
than  it  would  have  if  the  power  factor  were  unity. 

Tests  at  zero  power  factor,  on  the  other  hand,  have  the  advantage 
that  such  errors  are  insignificant;  furthermore,  the  tests  are  less  diffi- 
cult to  make  on  account  of  the  small  amount  of  power  required. 
They  can  often  be  made  when  tests  at  unity  power  factor  are  not 
possible. 

For  these  reasons,  specification  of  regulation  at  zero  power  factor 
(rather  than  unity  power  factor)  has  been  advocated;  such  specifica- 
tion can  be  checked  by  experiment  and,  furthermore,  it  gives  the 
regulation  under  the  worst  conditions.  On  the  other  hand,  this  is 
objected  to  because,  by  itself,  the  regulation  at  zero  power  factor  is 
no  positive  indication  of  the  behavior  of  the  machine  at  unity  power 
factor ;  two  machines  with  the  same  regulation  at  zero  power  factor 
may  have  very  different  regulations  at  unity  power  factor.  This  is 
largely  due  to  resistance  drop,  which  is  of  importance  at  unity  power 
factor,  but  has  practically  no  effect  at  zero  power  factor.  Specifica- 
tion of  regulation  at  zero  power  factor  is,  therefore,  insufficient — 


ioo  SYNCHRONOUS   ALTERNATORS.  [Exp. 

unless,  in  addition,  the  resistance  drop  is  separately  stated.  Tests  at 
zero  power  factor  are  also  objected  to  because  such  tests  are  made 
when  the  distorting  influence  of  cross-magnetization  is  absent.  (See 
Vol.  I.,  p.  761,  Int.  Elec.  Cong.,  1904.) 

§  55.  References. — References  are  given  below  to  a  few  leading 
articles  on  the  subject  of  alternator  regulation.  A  complete  list  would 
be  a  long  one,  but  the  references  here  given  are  the  best  ones  to  consult 
first;  they  contain  references  to  practically  all  that  has  been  written 
on  the  subject.  Rushmore's  paper,  with  twenty-four  references,  sum- 
marizes the  work  of  others  and  is  one  of  the  best  papers  to  read  first, 
particularly  in  connection  with  variations  of  the  magnetomotive  force 
and  electromotive  force  methods.  The  discussion,  found  at  the  close 
of  some  of  these  papers,  will  be  found  very  valuable. 

Transactions  International  Elect.  Congress,  St.  Louis,  1904: 
The  Regulation  of  Alternators,  by  D.  B.  Rushmore,  Vol.  I.,  p.  729; 
The/  Testing  of  Alternating  Current  Generators,  by  B.  A.  Behrend, 

Vol.  I.,  p.  528; 

Methods  of  Testing  Alternators  According  to  the  Theory  of  Two  Reac- 
tions, by  A.  Blondel,  Vol.  I.,  p.  620 ; 
Methods  of  Calculation  of  Armature  Reactions  of  Alternators,  by  A. 

Blondel,  Vol.  I.,  p.  635. 

Transactions  American  Inst.  of  Electrical  Engineers: 
The  Determination  of  Alternator  Characteristics,  by  L.  A.  Herdt,  Vol. 

XIX.,  p.  1093,  1902  ; 

The  Experimental  Basis  for  the  Theory  of  the  Regulation  of  Alter- 
nators, by  B.  A.  Behrend,  Vol.  XXL,  p.  497,  1903 ; 
A  Contribution  to  the  Theory  of  the  Regulation  of  Alternators,  by 

Hobart  and  Punga,  Vol.  XXIII.,  p.  291,  1904. 
Journal  British  Inst.  of  Electrical  Engineers: 

Henderson   and   Nicholson,   p.   465,    1905;    S.    P.    Smith,  paper   read 

November  12,  1908  (also  Lond.  Electrician,  November  13). 
See  also  Guilbert,  Elect.   World,  1902-3;  Torda-Heymann,  Lond.  Elec- 
trician, Vol.  LIIL,  p.  6,  1904. 

Parts  of  the  subject  will  be  found  treated  in  various  text-books:  S.  P. 
Thompson's  Dynamo  Electric  Machinery,  Karapetoff's  Exp.  Elect.  Eng., 
Franklin  and  Esty's  Elect.  Eng.,  etc. 


3-B]  PREDETERMINATION:  ;;VJ.  ]':  \\    \  /      101 

APPENDIX  III. 
MISCELLANEOUS   NOTES. 

§  56.  Transmission  Line  Regulation. — In  the  electromotive  force 
method,  §§  16-22,  a  complete  treatment  is  given  of  the  effect  upon 
delivered  voltage  of  resistance  drop  and  reactance  drop  in  the  arma- 
ture of  an  alternator.  The  treatment,  however,  is  general  and  is  not 
limited  to  alternators.  The  same  treatment  will  apply  to  any  resist- 
ance and  reactance  drop,  wherever  located,  and  may  accordingly  be 
applied  to  the  case  of  a  transmission  line.  In  the  geometrical  treat- 
ment of  any  problem,  resistance  drop  is  always  in  phase  with  the 
current,  reactance  drop  in  quadrature. 

Example  I. — Given  a  transmission  line  in  which  RI  drop  —  7.4;  XI 
drop  =  233.9.  What  must  be  the  voltage  Eo  applied  at  the  sending  end  of 
the  line  to  maintain  a  voltage  of  575  at  the  receiver  for  a  load  of  43.4 
amperes,  at  unity  power  factor,  at  power  factor  0.866  (current  lagging 
30°),  and  at  power  factor  0.866  (current  leading  30°)?  Figs.  3,  4  and  5 
show  that  627,  726  and  508  volts,  respectively,  are  required  at  the  sending 
end  in  the  three  cases,  the  corresponding  line  regulation  being  9,  26.3 
and  —  12  per  cent.  In  this  example  the  same  numerical  values  have  been 
used  for  a  transmission  line  as  were  used  in  Figs.  3,  4  and  5  for  an  alter- 
nator. Practical  values  for  a  transmission  line  would  give  a  relatively 
greater  resistance  drop  and  smaller  reactance  drop,  as  in  example  2. 

Example  2. — A  transmission  line  gives  1,000  volts  at  the  receiver.  The 
resistance  drop  is  100  volts,  reactance  drop  is  200  volts ;  what  is  the  regula- 
tion for  different  power  factors? 

Curves  as  shown  in  Figs.  6  and  7  can  be  drawn  for  a  transmission 
line.  These  curves  have  been  discussed  for  an  alternator;  the  discus- 
sion can,  however,  be  applied  to  a  transmission  line. 

In  calculating  the  regulation  of  a  transmission  line,  the  values  of 
resistance  and  reactance  can  be  taken  from  tables  in  various  hand- 
books and  elsewhere. 

In  testing  a  transmission  line,  the  reactance  drop  can  be  found  by 
an  open-circuit  test  and  a  short-circuit  test,  as  in  the  case  of  alterna- 
tors. With  a  low  voltage,  short-circuit  the  line  and  measure  7s ;  open- 
circuit  the  line  and  measure  EO.  The  line  impedance  is  Z  =  Eo  -=-  Is ; 
the  line  reactance  is  X  =  \JZZ  —  R2. 

In  the  laboratory  a  line  with  resistance  and  reactance  can  be  tested 
in  this  way  as  a  transmission  line;  the  regulation  for  loads  of  dif- 
ferent power  factors  can  be  predetermined  (Figs.  6  and  7)  and  com- 
pared with  actual  load  tests. 


CHAPTER    IV. 
SINGLE-PHASE  CURRENTS. 

EXPERIMENT  4-A.     Study  of  Series  and  Parallel  Circuits  Con- 
taining Resistance  and  Reactance. 

§  i.  Introductory. — The  object  of  this  experiment  is  to  acquaint 
one  with  the  fundamental  relations  between  currents  and  electro- 
motive forces  in  alternating  current  circuits.  These  relations  will 
be  brought  out  by  a  study  of  series  and  parallel  circuits  contain- 
ing resistance  and  inductance,  the  clear  understanding  of  which  is 
essential  for  one  undertaking  any  study  of  alternating  currents. 
Practically  every  problem  in  alternating  currents  involves — or  can 
be  reduced  to — a  problem  of  series  and  parallel  circuits.  A  study 
of  alternator  characteristics  (see  Figs.  3-5,  Exp.  3~B)  is  a  study 
of  series  circuits;  the  transformer  (see  Figs.  6-9,  Exp.  5~C)  can 
be  reduced  to  equivalent  series  and  parallel  circuits,  and  so,  too, 
the  induction  motor.  This  is  true  of  nearly  all  types  and  kinds 
of  alternating  current  apparatus.  It  will  be  found  that  the  study 
of  series  and  parallel  circuits  brings  out  the  general  principles 
that  are  common  to  all  alternating  current  problems.  Such  cir- 
cuits are  studied,  therefore,  not  merely  as  leading  up  to  the  sub- 
ject proper,  but  as  actually  being  the  subject  matter  of  all  alter- 
nating current  testing. 

Part  I.  contains  an  outline  of  the  underlying  principles  of  the 
subject,  which  will  be  found  discussed  in  detail  in  Bedell  and 
Crehore's  Alternating  Currents  and  in  other  treatises.  Part  II. 
describes  the  tests  to  be  made  and  Part  III.  describes  the  results 
derived  from  them.  For  the  convenience  of  the  reader,  some 
paragraphs  on  theory  are  included  in  Part  III. 

102 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  103 

PART    I.     ELEMENTARY    PRINCIPLES. 

§  2.  Defining  Relations. — In  a  direct  current  circuit,  the  cur- 
rent which  flows  is  I  =  E-7-R,  irrespective  of  whether  the  cir- 
cuit is  inductive  or  not;  the  power  expended  is  the  product  of 
electromotive  force  and  current. 

§  3.  In  a  non-inductive*  alternating  current  circuit,  this  is  also 
true;  the  current  is  determined  by  the  resistance,  as  in  a  direct 
current  circuit,  and  the  power  is  the  product  of  electromotive 
force  and  current;  thus, 

;    ,\  ~        ' 

The  impedance,  defined  below,  consists  in  this  case  of  the 
resistance  R  only. 

§4.  In  an  inductive  alternating  current  circuit,  the  current  is 
less  than  E-^-R  and  the  power  is  less  than  El',  thus, 

/  =  E  -f-  Z ;  W  =  EI  X  power  factor. 

The  impedance  Z,  defined  as  the  volts  per  ampere,  is  greater 
than  the  resistance  R  on  account  of  the  reactance  X ;  thus, 


The  reactance  (defined  in  §40)  for  an  inductive  circuit  has  a 
value  X  =  L<a,  where  L  is  the  inductance,  or  coefficient  of  self- 
induction  of  the  circuit,  and  <o  is  2?r  X  frequency  in  cycles  per 
second  (§  i,  Exp.  3~A).  Impedance  and  reactance  are  expressed 
in  ohms.  It  is  seen  that  inductive  reactancef  depends  not  only 

*  (§3a).  A  circuit  is  inductive  when  a  current  in  it  sets  up  a  magnetic 
field  (§  14)  ;  it  is  non-inductive  when  a  current  in  it  produces  no  magnetic 
field.  A  circuit  is  never  entirely  non-inductive,  but  may  be  made  nearly 
so.  This  is  practically  accomplished  when  the  outgoing  and  return  con- 
ductors are  placed  so  close  together  that  the  magnetic  effects  of  the 
currents  in  the  two  conductors  neutralize  each  other.  In  a  solenoid  this 
is  accomplished  by  using  a  double  winding,  the  currents  in  the  two  halves 
of  which  flow  in  opposite  directions. 

t  (§4a).  In  a  circuit  with  capacity  C,  the  reactance  is  \IC<a.  When  L 
and  C  are  both  present,  the  total  reactance  is  the  difference  between  the 
capacity  reactance  and  inductive  reactance;  X  =  L<a — i/Cw.  See  §57. 


104  SINGLE-PHASE  CURRENTS.  [Exp. 

upon  L,  which  is  a  constant  of  the  circuit  depending  upon  its 
form  and  dimensions,  but  also  upon  the  frequency  of  the  alter- 
nating current  supply. 

§  5.  The  preceding  equations  can  be  written 


VR2  +  X*     VR*  +  L2a> 


The  admittance  Y  of  an  alternating  current  circuit,  defined  as 
the  amperes  per  volt,  is  the  reciprocal  of  impedance;  Y  =  I-+-E. 
The  unit  of  admittance  is  commonly  called  the  mho. 

§  6.  Power  factor,  defined  as  the  ratio  of  true  power  W  to 
apparent  power  or  volt-amperes  El  }  is  always  less  than  (or  equal 
to)  unity.  Power  f  actor  =  cos  0,  where  0  is  the  phase  difference 
between  E  and  /  ;  see  Figs.  2  and  7  discussed  later.  In  a  circuit 
with  resistance  R  and  reactance  X, 


The  subject  will  be  most  readily  understood  by  considering: 
first,  circuits  with  R,  only;  second,  circuits  with  X,  only;  and 
finally  circuits  with  both  R  and  X. 

§  7.  Series  Circuit  with  Resistance  Only.  —  In  an  alternating 
current  circuit  "containing  only  a  resistance  R,  the  electromotive 
force  required  to  make  flow  a  current  /,  is 


as  in  a  direct  current  circuit. 

The  current  is  in  phase  with  the  electromotive  force.  As  the 
electromotive  force  rises  from  zero  to  a  maximum  and  falls  again 
to  zero,  the  current  i  at  each  instant  is  proportional  to  the  electro- 
motive force  e  at  that  instant;  e  =  Ri.  The  current  is  zero 
when  the  electromotive  force  is  zero,  and  is  a  maximum  when 
the  electromotive  force  is  a  maximum. 

§8.  If  E  is  represented  as  a  vector,  Fig.  5,  the  current  /  is 
represented  as  a  vector  in  the  same  direction  or  phase  as  E  ; 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  105 

I  that  is,  to  cause  a  current  /  to  flow  through  a  resistance  R,  an 
in-phase  electromotive  force  equal  to  RI  is  required. 
§  9.  Significance  of  Vectors. — In  developing  the  theory  of  vec- 
tor diagrams  for  alternating  current  quantities,  the  vectors  rep- 
resent the  maximum  values  of  quantities  which  vary  according 
to  a  sine  law.  In  applying  these  diagrams,  however,  the  vectors 
are  usually  drawn  to  represent  the  effective  (or  virtual)  values, 
as  measured  by  ammeter  and  voltmeter, — the  effective  value  of 
a  sine  wave  being  ^V2  times  its  maximum  value.*  Furthermore, 
vectors  are  used  for  currents  and  electromotive  forces  which  do 
not  vary  exactly  as  a  sine  law,  although  the  results  in  these  cases 
are  not,  in  general,  theoretically  correct.f  In  drawing  vector 
diagrams,  it  is  implied,  therefore,  that  the  currents  and  electro- 

I  motive  forces  have  wave  forms  which  are  sine  waves  or  may  be 
represented  by  equivalent  sine  waves  of  the  same  effective  values. 
The  phase  difference  0,  between  equivalent  sine  waves  for  cur- 
rent and  electromotive  •  force,  is  determined  by  the  relation: 
cos  0  =  power  f actor  =  W  -f-  EL 

§  10.  Direction  of  Rotation.  —  Counter-clockwise  rotation  is 
usually  taken  as  the  direction  of  rotation  of  alternating  current 
vector  diagrams,  and  this  convention  will  be  here  followed. 

By  considering  a  diagram  as  making  one  complete  revolution 
(360°)  in  one  cycle,  the  projections,  from  instant  to  instant,  of 
the  various  lines  of  the  diagram  upon  any  fixed  line  of  reference 
will  be  proportional  to  the  instantaneous  values  of  the  quantities 
represented  by  those  lines.  By  reversing -all  diagrams  as  in  a 
mirror,  the  corresponding  diagrams  for  clock-wise  rotation  will 
be  obtained. 

§11.  Electrical  Degrees. — In  alternating  current  vector  dia- 
grams, "  angle  "  is  a  measure  of  time,  360°  indicating  the  time 

*  See  Bedell  and  Crehore's  Alternating  Currents,  p.  38,  and  other  text- 
books. 

t  (§Qa).  Compare  §§60-64;  for  further  discussion,  see  references  given 
in  §  gb,  Exp.  5-C. 


106  SINGLE-PHASE  CURRENTS.  [Exp. 

of  one  complete  period  or  cycle,  90°  indicating  J  period,  etc. 
A  degree  is,  therefore,  a  unit  of  time,  being  sometimes  designated 
a  "time-degree"  or  "electrical  degree."  This  designation  is,  how- 
ever, unnecessary  except  in  discussions  where  "  space-degrees  " 
are  also  used. 

§  12.  Series  Circuit  with  Reactance  Only.  —  In  an  alternating 
current  circuit  containing  only  a  reactance  of  X  ohms,  the  electro- 
motive force  required  to  make  flow  a  current  I,  is 

EX  =  XI;    and  /  =  £x-5-X, 

as  shown  in  §§14-17. 

When  the  reactance  X,  is  due  to  inductance,  the  electromotive 
force  to  overcome  reactance  is 


Reactance  is  the  same  as  resistance  in  that  an  electromotive 
force  proportional  to  it  is  required  to  cause  a  current  to  flow, 
the  electromotive  force  being  XI  for  reactance 
and  RI  for  resistance.  Reactance  is,  however, 
different  from  resistance  in  that  it  consumes 
no  energy ;  when  the  current  is  increasing, 
energy  is  stored*  in  the  magnetic  field  (as  in 
a  fly-wheel),  this  energy  being  returned  to 
the  circuit  when  the  current  is  decreasing. 

In  a  reactance,  the  current  and  electromotive 
FIG.    i.       Vector 
diagram  for  circuit      force  are  not  in  phase  but  are  in  quadrature 

with   inductive   re-      with  each  other,  i.  e.,  the  current  and  electro- 

actance.  r  , .  ~,       .        t  «  ,. 

motive  force  differ  in  phase  by  a  quarter  of  a 
cycle  or  90°,  and  when  one  is  a  maximum  the  other  is  zero. 


*  (§  I2a).  The    energy   of   the   magnetic    field    is    equal   to    ^>L/2,    cor- 
responding to  the  energy  of  a  moving  body,  J/2MF*. 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  107 

§13.  For  inductive  reactance,*  the  electromotive  force  to  over- 
come reactance  is  in  advance  of  the  current  by  90°,  as  in  Fig.  I, 
and  is  not  in  phase  as  in  Fig.  5.  The  current  lags  behind  the 
electromotive  force  by  90°,  that  is,  the  current  reaches  a  positive 
maximum  J  cycle  later  than  the  electromotive  force  reaches  its 
positive  maximum.  When  R  =  o,  tan  6  =  X  -f-  R  =  oo  ;  #=90°  ; 
power  =  EI  cos  Q  =  o.  A  current  and  electromotive  force  in 
quadrature  represent  no  power  and  are  said  to  be  "  wattless." 

§  14.  Theory. — When  a  current  flows  in  an  inductive  circuit, 
the  current  sets  up  magnetic  flux  which  is  linked  with  the  circuit. 
When  the  current  changes,  this  flux  changes  and  a  counter- 
electromotive  force  is  induced  in  the  circuit  tending  to  oppose 
any  change  in  the  current, — the  current  seemingly  possessing 
inertia. 

The  electromotive  force  produced  by  self-induction  depends 
upon  the  rate  of  change  of  current,f  and  is 

e  oc  —  di/dt ;    or,    e  —  —  L  di/dt. 

The  negative  sign  indicates  that  the  electromotive  force  is  counter 
to  the  impressed  electromotive  force. 

The  equal  and  opposite  impressed  electromotive  force  to  over- 
come self-induction  is 

e  =  L  di/dt. 


*  (§  I3a).  For  capacity  reactance,  the  electromotive  force  to  overcome 
reactance  X  =  i/Cu  is  A7  =  I  -f-  Cw  and  is  90°  behind  the  current ;  the 
current  is  90°  in  advance  of  the  electromotive  force ;  see  §  55. 

t  (§  I4a).  The  electromotive  force  produced  by  self-induction,  expressed 
in  terms  of  rate  of  change  of  flux,  is  *  =  —  S  d<t>/dt,  '  (Compare  §§33, 
33a,  Exp.  5-A.)  In  the  absence  of  iron,  i  and  0  are  proportional  to  each 
other  and  L  is  constant.  In  this  case  Li  =  S<f>,  and  £  =  S<f>  -=-  i ';  ors  the 
inductance  of  a  coil  is  equal  to  the  flux-linkages  or  flux-turns  S<f>  for  unit 
current.  Since  <J>ccSi,  it  follows  that  L^S2,  other  things  (including 
dimensions  of  coil  and  leakage)  being  equal;  the  inductance  of  a  coil  is 
approximately  proportional  to  the  square  of  the  number  of  turns.  In  the 
presence  of  iron,  i  and  0  are  not  proportional,  and  L  is  not  constant  but 
varies  with  saturation. 


io8  SINGLE-PHASE  CURRENTS.  [Exp. 

§  15.  The  inductance  L  of  a  circuit  is  defined  by  the  foregoing 
equations.  When  e  is  in  volts  and  i  is  in  amperes,  L  is  in  henries. 
A  circuit  has  an  inductance  of  one  henry  when  a  change  of  cur- 
rent at  the  rate  of  one  ampere  per  second  induces  an  electro- 
motive force  of  one  volt. 

§  1 6.  When  the  current  varies  according  to  a  sine  law, 

i  =  /max  sin  w/. 
The  impressed  electromotive  force  is,  accordingly, 

e  =  L  di/dt  =  Lw/max  cos  o>t  =  La>/max  sin  ( at  +  90°  ) . 

The  impressed  electromotive  force  to  overcome  self-induction 
is,  therefore,  90°  in  advance  of  the  current;  the  current,  on  the 
other  hand,  lags  90°  behind  the  electromotive  force. 

§  17.  The  maximum  value  of  this  electromotive  force  is  seen 
to  be  Lw  times  the  maximum  value  of  the  current;  hence,  the 
effective  value  of  this  electromotive  force  is  Lo>  times  the  effective 
value  of  the  current,  that  is,  Ex  =  Lu>I  =  XI.  Fig.  I  and  the 
statements  in  §§  12,  13  are  thus  established. 

§  1 8.  Series  Circuit  with  Resistance  and  Inductive  React- 
ance.— In  a  circuit  with  both  R  and  X,  the  electromotive  force 
required  to  cause  a  current  /  to  flow  consists  of  two  components, 
which  have  been  separately  discussed  in  the  preceding  paragraphs  : 
RI,  in  phase  with  /,  to  overcome  resistance; 
XI,  90°  ahead  of  I,  to  overcome  reactance. 

Thus  in  Fig.  2,  if  OD  is  current,  OC  is  the  electromotive  force 
to  overcome  resistance  and  CA  is  the  electromotive  force  to  over- 
come* reactance,  OA  being  the  total  impressed  electromotive 
force.  These  electromotive  force  relations  are  fundamental  and 

*(§i8a).  These  electromotive  forces,  CA  and  OC  are  components  of 
the  impressed  electromotive  force.  In  the  opposite  sense,  as  counter- 
electromotive  forces,  we  have  the  counter-electromotive  force  AC,  lagging 
90°  behind  the  current,  produced  by  inductive  reactance ;  and,  the  counter- 
electromotive  force  CO,  opposite  in  phase  to  the  current,  produced  by 
resistance.  Compare  §15,  Exp.  6-A. 


4-A] 


SERIES  AND  PARALLEL  CIRCUITS. 


109 


are  shown  by  the  electromotive  force  triangle,  Fig.  2,  and  by  the 
following  equations : 


Lcol  = 

The  impedance  triangle,  Fig.  3,  is  derived  by  dividing  the  elec- 
tromotive forces,  Fig.  2,  by  /. 

§  19.  It  is  seen  that  the  electromotive  forces  XI  and  RI  are 
added  as  vectors.  If,  instead  of  a  single  X  and  R,  there  were 


Resistance 


FIG.  2.     Electromotive  force  triangle. 


0 


FIG.  3.    Impedance  triangle. 


several,  the  same  procedure  could  be  followed:  RJ,  RJ,  RJ, 
etc.,  Would  be  laid  off  in  phase  with  / ;  and  XJ,  XJ,  XJ,  etc.,  in 
quadrature  with  /. 

Electromotive  forces  in  a  series  circuit  are  added  as  vectors. 
Impedances,  resistances  and  reactances  in  a  series  circuit  are 
added  as  vectors. 

§  20.  The  total  drop  in  phase  with  /  is  2RI ',  the  total  drop  in 
quadrature  with  /  is  2X1.  Hence,  for  any  series  circuit, 


J,  and  Z  = 

The  total  resistance  of  a  series  circuit  is  seen  to  be  the  arith- 
metical sum  of  the  separate  resistances;  the  total  reactance  is  the 
arithmetical  sum  of  the  separate  reactances. 

For  further  discussion  of  series  circuits,  see  §§38-50;  for  par- 
allel circuits  see  §§  51-58- 


no  SINGLE-PHASE  CURRENTS.  [Exp. 

PART   II.     MEASUREMENTS. 

§21.  The  following  tests  require  a  resistance,  which  is  non- 
inductive  and  is  designated  Rl ;  and  a  coil,  which  is  inductive  and 
is  designated  R2L2.  It  is  desirable  to  have  the  resistance  and  the 
coil  take  currents  which  are  comparable  in  value  with  each  other, 
for  the  frequency  at  which  the  tests  are  made;  thus,  if  at  no 
volts,  60  cycles,  the  coil  takes  a  current  of  10  amperes,  the  resist- 
ance should  be  so  selected  that  at  no  volts  it  takes  a  current  of, 
say,  from  5  to  20  amperes.  Except  for  §  28,  the  coil  should  not 
have  an  iron  core,  so  that  there  are  no  losses  except  RI2. 

For  the  tests  of  §  26a  (which  may  precede  the  main  tests),  the 
windings  of  the  coil  should  be  divided  in  two  equal  parts,  which 
can  be  connected  in  series  and  in  parallel. 

§  22.  The  instruments  required  consist  of  a  voltmeter,  capable 
of  reading  the  supply  voltage  and  lower  voltages ;  an  ammeter 
capable  of  measuring  the  combined  currents  of  the  coil  and  resist- 
ance; and  a  wattmeter  having  a  voltage  range  corresponding  to 
the  range  of  the  voltmeter  and  a  current  range  corresponding  to 
the  range  of  the  ammeter. 

A  voltmeter  switch  will  be  found  convenient  for  the  series  tests 
( §§  29~3J )  and  an  ammeter  switch  for  the  parallel  tests  ( §§  32-34) . 

On  all  tests  the  frequency  should  be  known. 

§23.  (a)  Resistance  Alone. — With  an  adjusting  resistance  in 
series,  as  in  Fig.  4,  connect  the  resistance  Rl  to  the  supply  circuit 
(say  no  volts,  60  cycles)  and  measure  the  current  I,  the  voltage 
E  at  the  terminals  of  Rlt  and  the  watts  W  consumed  by  Rlf  The 
current  coil  of  the  wattmeter  is  connected  in  series  as  an  ammeter 
and  the  potential  coil  in  shunt  as  a  voltmeter,  the  arrangement* 
of  instruments  being  shown  in  Fig.  i,  Exp.  5-B. 

*  (§23a).  In  these  tests  no  account  is  ordinarily  to  be  taken  of  the  fact 
that  the  instruments  themselves  consume  a  certain  small  amount  of 
power,  as  fully  discussed  in  Appendix  III.,  Exp.  S-A ;  this  fact,  however, 
should  not  be  neglected  in  accurate  testing,  as  for  example  in  the  accurate 
determination  of  L  by  the  impedance  method,  §  47. 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  m 

§24.  Vary  the  adjusting  resistance,*  and  in  this  way  take  sev- 
eral sets  of  readings. 

If  there  is  any  question  as  to  the  accuracy  of  the  instruments, 
assume  the  ammeter  and  voltmeter  to  be  correct  and  determine  a 
correction  for  the  wattmeter,  so  that  in  (a)  the  watts  as  read  by 
the  wattmeter  are  equal  to  the  product  of  volts  and  amperes,  as 
read  by  the  voltmeter  and  ammeter.  This  serves  as  a  calibration 
of  the  wattmeter,  to  be  used  in  this  and  subsequent  tests. 

§  25.  Take  readings,  in  a  like  manner,  at  a  second  frequency. 

§26.  (&)  Coil  Alone.f — Repeat  (a)  using  the  coil  R2L2  alone, 
as  in  Fig.  6,  instead  of  the  resistance  R^. 

§  27.  Take  readings  at  a  second  frequency. 

§  28.  Effect  of  Iron. — Gradually  introduce  an  iron  core  and 
watch  the  ammeter ;  or,  introduce  iron  wires,  a  few  at  a  time,  thus 
gradually  increasing  the  amount  of  iron.  At  present,  only  the 
general  effect  of  iron  is  to  be  noted  and  explained;  a  more  com- 
plete study  of  iron  in  the  form  of  a  closed  magnetic  circuit  is 
made  in  the  subsequent  experiments  on  the  transformer. 

§29.  (c)  Resistance  and  Coil  in  Series. — Connect  the  resist- 
ance Rl  and  the  coil  R2L2  in  series,  and,  together  with  an  adjust- 
ing resistance,  connect  to  the  supply,  as  in  Fig.  8.  For  a  certain 
current,  take  readings  of  the  voltage  drop,  the  current  and  the 
watts  consumed  as  follows:  first,  for  the  resistance;  second,  for 

*  (§24a).  This  adjustment  should  be  so  made  that  the  readings  of  the 
various  instruments  are  taken  at  open  parts  of  the  scales. 

t(§26a).  Series  and  Parallel  Connections. — If  is  instructive  to  use  a 
coil  with  two  equal  windings.  In  this  case,  the  regular  tests  should  be 
made  with  the  two  windings  either  in  parallel  or  in  series  and  additive, — 
i.  e.,  setting  up  magnetic  flux  in  the  same  direction.  If  one  winding  is 
reversed,  it  will  oppose  the  other  so  that  the  resultant  flux  (and  hence 
the  impedance)  is  small.  A  few  volts  may  cause  a  very  large  current. 

Preliminary  Test. — With  the  resistance  R-L  in  series  as  a  safeguard,  to 
avoid  excessive  current,  measure  the  current  and  voltage  and  determine 
the  impedance  of  each  winding  alone  and  of  the  two  windings  connected 
in  series  and  in  parallel,  additively  and  differentially.  The  additive  wind- 
ing is  inductive;  the  differential  winding  is  non-inductive, — except  so  far 
as  there  is  magnetic  leakage. 


H2  SINGLE-PHASE  CURRENTS.  [Exp. 

the  coil;  and  third,  for  the  resistance  and  coil  combined.  Vary 
the  current,  by  means  of  the  adjusting  resistance,  and  take  several 
sets  of  readings,  the  current  being  kept  constant  for  each  set; 
see  §24a. 

§  30.  The  ammeter  and  current  coil  of  the  wattmeter  are  in 
series  with  the  circuit  for  all  readings  and  their  location  is  un- 
changed. The  voltmeter  and  the  voltage  coil  of  the  wattmeter 
are  in  parallel  with  each  other  and  are  connected :  first,  across  the 
terminals  of  the  resistance ;  second,  across  the  terminals  of  the 
coil;  and  third,  across  the  terminals  of  the  resistance  and  coil 
combined.  These  changes  can  be  most  readily  made  by  means  of 
a  voltmeter  switch,  the  current  being  maintained  constant  during 
one  set  of  readings  by  means  of  the  adjusting  resistance.  Some 
error  is  here  introduced  on  account  of  the  power  consumed  in  the 
instruments. 

§31.  Repeat  at  a  second  frequency. 

§32.  (d)  Resistance  and  Coil  in  Parallel. — Connect  the  resist- 
ance R±  and  the  coil  R2L2  in '  parallel,  and,  together  with  the 
adjusting  resistance,  connect  to  the  supply  as  in  Fig.  10.  For 
a  certain  constant  voltage  E,  take  readings  of  current,  voltage 
and  watts:  first,  for  the  resistance  alone;  second,  for  the 
coil  alone;  and  third,  for  the  resistance  and  coil  together  in 
parallel. 

Vary  the  voltage  by  means  of  the  adjusting  resistance,  and  take 
several  sets  of  readings,  the  voltage  being  kept  constant  for  each 
set ;  see  §  24a. 

§  33.  The  voltmeter  and  potential  coil  of  the  wattmeter  are  not 
changed  during  the  readings.  The  ammeter  and  the  current  coil 
of  the  wattmeter  are  shifted  from  one  circuit  to  another,  being: 
first,  in  series  with  the  resistance ;  second,  in  series  with  the  coil  ; 
and  third,  in  the  main  circuit.  Since,  during  one  set  of  readings, 
the  voltage  is  maintained  constant,  the  readings  thus  obtained* 

*This  would  be  true  if  the  instruments  themselves  took  no  power; 
§  233. 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  ^3 

will  be  the  same  as  readings  obtained  simultaneously  with  three 
ammeters  and  three  wattmeters. 

§  34.  Repeat  at  a  second  frequency. 

§  35.  (e)  Measurement  of  Resistance. — Measure  the  resist- 
ances R-L  and  R2  by  direct  current,  §  17,  Exp.  i-A. 

PART  III.  RESULTS. 

§36.  In  each  test,  (a),  (b),  (c),  and  (d),  select  say  two  sets 
of  readings  at  each  frequency  and  construct  vector  diagrams 
showing  the  magnitude  and  relative  phase  positions  of  the  various 
currents  and  voltages.  Compute  for  the  various  circuits,  and 
parts  of  circuits,  the  power  factor  and  the  phase  difference 
between  current  and  voltage.  The  prime  object  is  to  obtain  a 
clear  understanding  of  the  relations  between  the  various  quanti- 
ties, rather  than  to  obtain  exact  numerical  values. 

Adjusting 
Resistance 


I  FIG.  4. 

Circuit  containing  resistance 
§37.   (a)  Resistance  Alone. — For  this  case,  the  current  and 
electromotive   force  are  in  phase,  and  true  power  is   equal  to 
the  product,  volts   X   amperes.      Power  f actor  =  W  -=-  EI=  I ; 
cos  9=  i ;    0  =  o.     See  Fig.  5. 

§38.   (b)  Coil  Alone. — The  current  I  lags  behind  the  electro- 
motive force  E  by  an  angle  0,  as  in  Fig.  7.     The  true  power  W, 
indicated   by   the   wattmeter,   is   less   than   the   volt-amperes   or 
apparent  power,  El ;  thus 
9 


Hence 


SINGLE-PHASE  CURRENTS. 

W  =  El  X  power  factor  =  El  cos  6. 

cos  0  =  power  factor  =  W  -=-  EL 


[Exp. 


The  angle  0  is  thus  computed  from  the  readings  of  the  wattmeter, 
voltmeter  and  ammeter. 

In  constructing  Fig.  7,  lay  off  OA  =  E ;  then  lay  off  OD  =  I, 
at  an  angle  6  determined  as  above,  and  construct  the  right  tri- 
angle of  electromotive  forces,  OCA. 

Adjusting 
Resistance 


-w^yvj" 

J 

1 

1 

R2 
L2 

j 

T 

FIG.  6. 


Circuit  containing  coil  R2L2. 


Compute  the  components  of  electromotive  force  and  current, 
and  verify  the  various  relations  discussed  in  the  following 
paragraphs. 

§39.  Components  of  Electromotive  Force. — In  the  manner  just 
described,  the  electromotive  force  is  resolved  into  the  power  com- 
ponent, Ep  =  OC,  in  phase  with  /,  and  the  wattless  component, 
EQ  =  CA,  in  quadrature  with  /.  These  components  are 

Ep  =  E  cos  0  =  E  X  power  factor ; 
EQ  =  E  s'mO  =  E  X  reactive  factor.* 

The  vector  sum  of  these  two  components  gives  the  total  im- 
pressed electromotive  force. 


*  (§39a).  Designating  power  factor  by  p  and  reactive  factor  by  q,  it  is 
seen  that  />2  +  #2=i.     Compare  Standardization  Rule  56. 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  115 

§  40.  From  these  electromotive  forces,  we  have  the  definitions  : 

Impedance  is  total  electromotive  force  divided  by  current; 
Z  =  E-+-L 

Resistance  is  power  or  in-phase  component  of  electromotive 
force  divided  by  current;  R  =  Ecos0^r-L  (In  general,  when 
motors,  transformers,  etc.,  are  included  in  the  circuit,  this  gives 
apparent  resistance.) 

Reactance*  is  the  wattless  or  quadrature  component  of  electro- 
motive force  divided  by  current;  X  =  Esm0-^I. 

§41.  Components  of  Current.^  —  In  a  similar  manner,  the  cur- 
rent may  be  resolved  into  a  power  component,  /P  =  /cos0,  in 
phase  with  E,  and  a  wattless  component  /g  =  /sin0,  in  quadra- 
ture with  E;  the  total  current  is  I=^/Ip2  -}-  IQZ. 

§  42.  From  these  currents,  we  have  the  definitions  :  . 

Admittance  Y  is  total  current  divided  by  electromotive  force; 
K=/-*-E. 

Conductance  g  is  the  power  or  in-phase  component  of  current 
divided  by  electromotive  force;  g  =  IcosQ^-E. 

Susceptance  b  is  the  wattless  or  quadrature  component  of  cur- 
rent divided  by  electromotive  force;  b  =  I  sinO-^-E. 

We  have,  then,  the  following  relations; 

Total  current  =  I  =  E  X  Y. 
Power  current  =  /  cos  6  =  E  X  9> 
Wattless  current  =  /  sin  0  =  E  X  b. 
g=Ycos6. 


Admittance  =  V^2  + 


*  This  is  the  general  definition,  Lta,  i/Cw,  etc.,  being  merely  particular 
values;  see  paper  on  Reactance,  by  Steinmetz  and  Bedell,  p.  640,  Vol. 
XI.,  Transactions  A.  I.  E.  E.,  1894. 

f  (§4ia).  As  an  illustration  of  the  resolution  of  current,  see  Fig.  2  and 
other  figures 'in  Exp.  5-C.  It  is  usual  to  resolve  electromotive  force  into 
components  for  series  circuits  and  current  into  components  for  parallel 
circuits. 


n6  SINGLE-PHASE  CURRENTS.  [Exp. 

Admittance  is  the  reciprocal  of  impedance;  but  conductance  is 
not  the  reciprocal  of  resistance  (as  with  direct  currents),  nor  is 
susceptance  the  reciprocal  of  reactance. 

§43.  Power. — It  is  seen  that  the  expression  for  true  power, 
El  cos  0,  may  be  written  in  two  ways: 

W  =  Ecos6y<  I  (resolving  electromotive  force); 
or, 

£F=/cos#XE   (resolving  current). 

§44.  Resolving  the  electromotive  force  into  components,  we 
have:  True  power  is  equal  to  the  product  of  current  (I)  and 
the  component  of  electromotive  force  (EcosO)  which  is  in  phase 
with  the  current. 

§  45.  Resolving  the  current  into  components,  we  have :  True 
power  is  equal  to  the  product  of  the  impressed  electromotive 
force  (E)  and  the  component  of  current  (I  cos  6)  which  is  in 
phase  with  the  electromotive  force. 

§46.  Calculation  of  L  and  X  by  Wattmeter  Method. — React- 
ance is  by  definition  (§40)  equal  to  the  quadrature  electromotive 
force,  Ex,  divided  by  current. 

Referring  to  Fig.  7,  the  reactance  and  inductance  of  the  coil 
R2L2  are  computed  as  follows: 

L2w  ^X2  =  CA  -r- 1,  ohms ; 
L2  =  X2  -f-  w  =  X2  -+-  2-n-n,  henries. 

By  this  method,  X2  and  L2  are  determined  by  measurements 
of  E,  I  and  W ' ,  and  are  independent  of  the  measured  value  of  R2. 
(See  §§47  and  49.)  Note  also  that  R2=  OC-t-I=  W-^-I2, 
and  that  tan  6  =  X2-^-R2. 

§  47.  Calculation  of  L  and  X  by  Impedance  Method. — By  the 
impedance  method,  L2  depends  upon  E,  I  and  the  measured  value 
of  R2,  and  is  independent  of  the  wattmeter  reading.  The  cal- 
culations are  made  as  follows: 

Impedance   ( ohms )  =Z2  =  E-^-I. 


Reactance   (ohms)  =X2=  \/Z22  —  R*. 


4-A] 


SERIES  AND  PARALLEL  CIRCUITS. 


117 


Here  R2  is  the  resistance  of  the  coil,  as  measured  by  direct 
current.  The  inductance,  in  henries,  is  L2  =  X2-^~2Trn. 

For  the  accurate  determination  of  L  by  either  of  these  methods, 
the  wave  form  of  electromotive  force  should  be  sinusoidal  and 
the  losses  in  instruments  should  be  taken  into  consideration,  §  23a. 

§48.  (c)  Resistance  and  Coil  in  Series. — In  a  series  circuit 
there  is  one  current  which  is  the  same  in  all  parts  of  the  circuit; 
electromotive  forces  are  added  vectorially,  i.  e.,  the  voltage  drops 
around  the  separate  parts  of  the  circuit,  when  added  as  vectors, 
give  the  total  impressed  electromotive  force  of  the  circuit. 


FIG.  8. 


O      /       D 

Resistance  and  coil  in  series. 


The  three  readings  of  the  voltmeter,  E,  E^  and  E2,  are,  accord- 
ingly, drawn  to  scale  so  as  to  form  the  triangle  OAB,  in  Fig.  9. 
The  current  /  is  laid  off  in  phase  with  Elt  since  the  current  and 
electromotive  force  in  the  non-inductive  resistance  are  in  the  same 
phase.  OCA  is  then  drawn  as  a  right  triangle. 

We  have  then  the  in-phase,  electromotive  forces,  OB  =  RJ  to 
overcome  the  resistance  R^  and  BC  =  R2I  to  overcome  the  resist- 
ance R2 ;  and  the  quadrature  electromotive  force,  CA  =  L2ul 
=  X2I,  to  overcome  the  reactance  X2.  It  will  be  seen  that  Fig.  9 
is  the  same  as  Figs.  5  and  7  combined  in  one  diagram  so  drawn 
that  the  current  in  both  parts  of  the  circuit  is  the  same  in  magni- 
tude and  in  phase. 

§49.  Three-voltmeter  Method. — The  foregoing  construction, 
known  as  the  three-voltmeter  method,  enables  us  to  calculate  L2 


n8  SINGLE-PHASE  CURRENTS.  [Exp. 

and  X2,  the  results  being  dependent  upon  three  voltmeter  read- 
ings and  current,  and  not  dependent  upon  the  wattmeter  (as  in 
the  wattmeter  method,  §46),  nor  upon  the  measurement  of  resist- 
ance (as  in  the  impedance  method,  §47). 
Referring  to  Fig.  9,  we  have 


hence 
and 


In  applying  the  three-voltmeter  method,  greatest  accuracy  is 
obtained  when  £1  =  £2.  If  an  electrostatic  voltmeter  is  used,  no 
error  is  introduced  on  account  of  power  consumed  in  the  instru- 
ment, §  23a. 

§  50.  Three-voltmeter  Method  for  Measuring  Power.  —  Before  the 
perfection  and  general  introduction  of  the  wattmeter,  the  three-volt- 
meter method  for  measuring  power  was  used  ;  this  is  now  obsolete  for 
practical  testing.  The  procedure  was  as  follows: 

Given  a  device  R^L2  (which  might  be,  for  example,  a  transformer) 
the  power  in  which  is  to  be  measured.  Connect*  in  series  a  non- 
inductive  resistance  -Rlf  as  in  Fig.  8,  and  read  E,  Ev  E2  and  /.  The 
power  in  R3L2  (see  Fig.  9)  is 

Wz  =  EJ  cos  02  =  (I  -r-  2EJ  (E*  -  E*  -  £22). 

See  Bedell  and  Crehore's  Alternating  Currents,  p.  232.  The  weak 
point  in  the  method  is  that  small  errors  in  observation  make  large 
errors  in  the  result.  The  three-ammeter  method,  with  a  non-inductive 
resistance  in  parallel  with  the  apparatus  under  test  as  in  Fig.  10,  is 
open  to  the  same  objection. 

§51.  (d)  Resistance  and  Coil  in  Parallel.  —  In  parallel  cir- 
cuits,* currents  combine  vectorially,  the  main  current  being  the 
vector  sum  of  the  branch  currents. 

*  (§5ia).  Currents  are  proportional  to  admittances;  hence  admittances 
may  be  added  as  vectors.  The  admittance  of  several  circuits  in  parallel 


4-A] 


SERIES  AND  PARALLEL  CIRCUITS. 


The  main  current  /  is  laid  off,  in  Fig.  n,  as  the  diagonal  of  a 
parallelogram  with  sides  equal  to  the  branch  currents  7X  and  /2. 
The  electromotive  force  E  is  laid  off  in  the  direction  of  Ilf  since 


Adjusting 
Resistance 


E 


FIG.  10. 


O     Iz          #2/2  C 
FIG.  ii. 


Resistance  and  coil  in  parallel. 


the  current  and  electromotive  force  in  the  non-inductive  branch 
are  in  phase.  E  is  the  common  terminal  electromotive  force  and 
is  the  same  for  both  branches. 

For  the  inductive  branch,  the  electromotive  force  triangle  OCA 
is  constructed,  as  in  (&).  For  this  branch,  the  power  electro- 
motive force  is-OC,  in  phase  with  /2;  the  wattless  electromotive 
force  is  CA  in  quadrature  with  /2.  Fig.  1  1  is  seen  to  be  the  same 
as  Figs.  5  and  7,  drawn  with  a  common  E  and  combined.  In 
Fig.  9,  these  figures  were  combined  with  a  common  I. 

§  52.  The  right  triangle  OC'A  is  the  electromotive  force  tri- 
angle for  an  equivalent*  single  circuit,  R'L',  which  could  be  sub- 
stituted for  the  two  parallel  circuits.  Since  OC'  =  R'I,  and 

is  the  vector  sum  of  the  admittance  of  the  separate  branches.  In  parallel 
circuits  we  may  add  as  vectors  either  currents  or  admittances;  while  in 
series  circuits  we  may  add  as  vectors  either  electromotive  forces  or  im- 
pedances, §§19  and  20. 

*  (§52a).  For  a  more  complete  discussion  of  equivalent  resistance  and 
inductance,  see  Bedell  and  Crehore's  Alternating  Currents,  pp.  238-41. 
Both  R'  and  L'  depend  upon  frequency  and  are  not  constants  of  the 
circuits;  the  equivalent  resistance  of  parallel  circuits  is  not  the  same  for 
alternating  as  for  direct  current. 


120  SINGLE-PHASE  CURRENTS.  [Exp. 

C'A=L'uI,  the  resistance  and  reactance  of  this  equivalent  cir- 
cuit are  computed  as  follows  : 


L'<»=CA~I. 

§  53.  For  any  number  of  parallel  circuits,  the  total  current  in 
phase  with  E  is  2,1  cos  0  ;  the  total  quadrature  current  is  2/  sin  0. 
Hence 


1=  V  (2/cos0)2+  (S/  sin  0)2. 
Dividing  by  E,  we  have 


The  total  conductance  of  a  number  of  parallel  circuits  is  the 
arithmetical  sum  of  the  separate  conductances;  the  total  suscept- 
ance  is  the  arithmetical  sum  of  the  separate  susceptances.  (Com- 
pare with  §20  for  series  circuits.) 


APPENDIX   I. 
CIRCUITS   WITH   CAPACITY. 

§  54.  It  is  not  intended  in  this  experiment  that  tests  with  capacity 
be  included,  the  following  summarized  statements  concerning  capacity 
being  made  for  reference  and  for  comparison  with  the  relations 
already  discussed  concerning  inductance. 

§  55.  Circuits  with  Resistance  and  Capacity. — In  theory,  circuits 
containing  capacity  (C)  can  be  treated  exactly  the  same  as  circuits 
containing  inductance,  if  the  following  differences  are  noted: 

Inductive  reactance  =  Leo ;  current  lags  behind  impressed  electro- 
motive force. 

Capacity  reactance  =  i  -=-  Cw ;  current  is  in  advance  of  impressed 
electromotive  force. 

In  either  circuit,  tan  0  =  X  -f-  R. 

All  diagrams  for  inductive  circuits  can  be  applied  to  capacity  cir- 


4-A]  SERIES  AND  PARALLEL  CIRCUITS.  121 

cults  by  writing  i  -=-  C<o  in  place  of  Lw,  and  reversing  the  diagrams 
(as  in  a  mirror)  so  that  current  is  leading  instead  of  lagging. 

Inductance  produces  effects  similar  to  mass  in  a  moving  mech- 
anism ;  capacity  produces  effects  similar  to  a  spring.  Inductance  and 
capacity  store,  but  do  not  consume,*  energy;  the  stored  energy  being 
JL/2  in  inductance  and  JC£2  in  capacity. 

§  56.  As  frequency  is  increased,  the  impedance  of  an  inductive  cir- 

icuit  becomes  greater;  the  impedance  of  a  capacity  circuit  becomes 
less.  Furthermore,  as  frequency  is  increased,  0  in  an  inductive  circuit 
becomes  greater;  0  in  a  capacity  circuit  becomes  less. 

§  57.  Circuits  with  Inductance  and  Capacity. — When  a  circuit  con- 
tains both  inductance  and  capacity,  the  total  reactance  of  the  circuit 
is  the  difference  between  the  inductive  reactance  and  the  capacity 
reactance ;  X  =  Loo  —  i/Cco.  Inductance  and  capacity  tend  to  neu- 
tralize each  other.  When  the  inductive  reactance  is  greater  than  the 
capacity  reactance,  the  current  lags  behind  the  electromotive  force, 
as  in  an  inductive  circuit ;  when,  on  the  other  hand,  the  capacity  react- 
ance is  the  greater,  the  current  is  in  advance  of  the  electromotive 
force,  as  in  a  capacity  circuit.  In  either  case,  tsn\0  =  X-^-R. 

§  58.  Voltage  Resonance. — In  a  series  circuit,  the  total  impedance 
may,  therefore,  be  less  than  the  impedance  of  part  of  the  circuit  only, 
and  the  total  impressed  voltage  may,  accordingly,  be  less  than  the 
voltage  drop  around  part  of  the  circuit  only.  The  voltage  around 
part  of  the  circuit  is  thus  increased  by  resonance  so  as  to  be  greater 
than  the  impressed  electromotive  force. 

§  59.  Current  Resonance. — In  parallel  circuits  with  two  branches, 
one  with  inductance  and  the  other  with  capacity,  the  current  in  the 
inductance  branch  is  lagging  while  the  current  in  the  capacity  branch 
is  leading.  The  two  branch  currents  are  to  a 'certain  extent  in  phase 
opposition  so  that  the  total  or  main  line  current,  which  is  the  vector 
sum  of  the  two,  may  be  less  than  the  current  in  either  branch. 
Due  to  resonance,  a  local  circulating  current  is  obtained  which  is 
greater  than  the  current  from  the  generator. 

§  60.  Non-Sine  Waves. — When  the  impressed  electromotive  force 
does  not  follow  a  sine  law,  there  are  present — in  addition  to  the 

*  In  an  inductance  with  iron,  some  energy  is  lost  in  magnetic  hysteresis ; 
similarly,  in  a  condenser,  a  small  amount  of  energy  is  lost  in  dielectric 
hysteresis. 


122  SINGLE-PHASE  CURRENTS.  [Exp. 

fundamental  —  harmonics  with  frequencies  higher  than  the  fun- 
damental. 

§  61.  In  inductive  circuits,  reactance  increases  with  frequency. 
Inductive  reactance  is,  accordingly,  greater  for  these  harmonics  than 
for  the  fundamental ;  harmonics  in  the  current  are  choked  out  and 
the  current  wave  is  more  nearly  a  sine  wave  than  is  the  wave  of 
impressed  electromotive  force.  For  this  reason,  vector  diagrams 
can  be  used  for  representing  experimental  results,  obtained  from 
measurements  on  inductive  circuits,  without  much  error — even  when 
the  impressed  electromotive  force  is  not  a  true  sine  wave. 

§  62.  Capacity  reactance,  on  the  other  hand,  decreases  with  fre- 
quency and  harmonics  in  the  current  are,  accordingly,  augmented; 
hence,  a  small  distortion  in  the  electromotive  force  wave  may  make 
a  large  distortion  in  the  current  wave. 

§  63.  For  this  reason,  vector  diagrams  are  less  accurate  for  rep- 
resenting experimental  results  for  capacity  circuits  than  for  inductive 
circuits,  when  the  electromotive  force  is  not  a  true  sine  wave.  For 
example,  in  the  laboratory  without  special  precautions,  capacity  react- 
ance can  not  be  measured  by  alternating  current  methods  as  accu- 
rately as  inductive  reactance. 

§  64.  When  capacity  and  inductance  are  both  present,  a  small  dis- 
tortion in  the  electromotive  force  wave  may  be  much  exaggerated 
by  the  resonance  of  a  particular  harmonic,  with  corresponding  error 
in  any  vector  representation. 


4-B]  CIRCLE  DIAGRAM.  123 

EXPERIMENT  4-B.     Circle  Diagram  for  a  Circuit  with  Resist- 
ance and  Reactance. 

§  I.  Introductory. — If  a  circuit  with  resistance  and  reactance 
is  supplied  with  a  constant  impressed  electromotive  force,  the 
current  will  have  a  certain  value  and  a  certain  phase  position 
with  reference  to  the  electromotive  force,  as  discussed  in 
Exp.  4-A. 


These  values  of  current  and  phase  angle  will  be  changed,  if 
either  the  resistance  or  the  reactance  of  the  circuit  is  changed. 

§  2.  In  a  circuit  in  which  the  reactance  X  is  constant,  and  the 
resistance  R  is  varied,  the  value  of  /  and  0  will  increase  when  R 
is  decreased;  as  resistance  is  cut  out  of  circuit,  the  current  will, 
accordingly,  not  only  be  larger  but  will  be  more  out  of  phase  with 
reference  to  the  electromotive  force.  In  the  limiting  cases :  when 
R  =  o,  the  current  is  E-^-X  and  (in  the  case  of  inductive  react- 
ance) lags  90°  behind  the  electromotive  force;  when  ^=00, 
7=^0,  and  6  =  0. 

§  3.  The  object  of  this  experiment  is  to  show  the  change  of  cur- 
rent in  magnitude  and  phase,  in  a  circuit  with  constant  inductive* 
reactance,  when  the  resistance  is  varied  and  the  impressed  elec- 
tromotive force  is  maintained  constant.  It  will  be  found  that  the 
locusf  of  the  current  vector  is  the  arc  of  a  semicircle,  as  in 
Fig.  2 ;  this  is  true  of  any  constant  potential  circuit,  in  which  the 
reactance  is  constant  and  the  power  consumption  is  variable — as 
in  a  transformer  (Exp.  5~C)  or  in  an  induction  motor. 

*  (§3a).  A  similar  experiment  may  be  performed  with  capacity  react- 
ance ;  see  Appendix  I.,  Exp.  4-A. 

A  converse  experiment  may  also  be  made  with  constant  resistance  and 
variable  reactance,  in  which  case  the  diameter  of  the  semi-circle  locus  is 
in  the  direction  of  E,  instead  of  at  right  angles  to  it;  see  reference,  §  3b. 

t(§3b).  Established  by  Bedell  and  Crehore,  Alternating  Currents,  pp. 
223  and  275. 


I24  SINGLE-PHASE  CURRENTS.  [Exp. 

§4.  Data. — Let  the  circuit  be  as  shown  in  Fig.  8,  Exp.  4-A, 
in  which  R^  is  a  non-inductive  resistance  and  R2L2  is  a  coil  (with- 
out iron)  with  resistance  R2,  inductance  L2  and  reactance  X2. 
The  impressed  electromotive  force  E  should  be  constant;  in  case 
E  varies,  all  readings  should  be  reduced  by  direct  proportion  to 
correspond  to  some  constant  value  of  E ;  an  adjusting  resist- 
ance, as  shown  in  Fig.  8,  Exp.  4-A,  is  unnecessary.  The  fre- 
quency should  be  constant. 

With  an  ammeter,  measure  the  current  7.  With  a  voltmeter, 
measure  the  various  falls  of  potential  as  follows :  E,  the  im- 
pressed electromotive  force;  Eit  the  fall  of  potential  around 
the  non-inductive  resistance  R^;  E2,  the  fall  around  the  coil 
RZL2. 

The  error  due  to  the  current  taken  by  the  voltmeter,  although 
negligible  for  a  circuit  in  which  .the  current  is  large,  becomes 
appreciable  when  the  current  is  small ;  this  error  may  be  avoided 
by  using  an  electrostatic  voltmeter,  which  takes  only  sufficient 
current,  to  charge  the  instrument. 

§  5.  Take  a  series  of  readings  for  decreasing  values  of  R2 
throughout  the  range  that  it  is  possible  to  read  E1  and  E2. 

§  6.  Repeat  at  a  second  frequency. 

§  7.  Repeat  at  one  frequency  with  an  iron  core  in  the  coil. 

§  8.  Measure  the  resistance  of  the  coil,  R2,  by  direct  current, 
fall-of-potential  method,  §17,  Exp.  i-A. 

§9.  Results. — For  one  set  of  readings,  draw  a  triangle,  OAB, 
Fig.  i,  with  the  observed  values  of  E,  E^  and£2  as  the  three  sides. 
Lay  off  OD  in  the  direction  of  OB,  equal  to  the  current  /,  in  any 
convenient  scale.  Produce  OB  to  C  by  an  amount  BC  =  R2I,  the 
electromotive  force  to  overcome  the  resistance  and  supply  the 
RI2  losses  in  the  coil.  OC  is  the  electromotive  force  to  over- 
come the  resistance  of  the  entire  circuit.  The  current  and  elec- 
tromotive forces  are  now  represented  in  magnitude  and  direction 
for  one  value  of  the  resistance.  Fig.  I  is  the  typical  diagram  for 


4-BJ 


CIRCLE  DIAGRAM. 


12 


a  series  circuit,  being  the  same  as  Fig.  9,  Exp.  4-A,  and  Fig.  3, 
Exp.  3~B,  for  the  alternator;  compare,  also,  Fig.  9,  Exp.  5-B 
and  the  transformer  diagrams,  Exp.  5~C.  As  explained  in  §  n, 
OCA  is  not  an  exact  right  angle. 

§  10.  For  a  second  set  of  readings,  locate  the  points  B',  C',  D', 
in  the  same  way  as  the  points  B,  C,  D  were  located. 

Locate  points  in  this  manner  for  all  the  readings,  thus  defining 
the  curves  in  Fig.  2,  which  are  the  loci  of  the  points  B,  C  and  D. 


Method  of  plotting 
results. 


FIG.  2.  Circle  diagram  for  a  constant  po- 
tential circuit  with  constant  reactance,  when 
resistance  is  varied.. 


§  ii.  It  is  seen  that,  as  the  resistance  is  decreased,  the  cur- 
rent increases  and  lags  more  and  more  behind  the  electromotive 
force.  If  the  impressed  electromotive  force,  and  hence  the  current 
which  flows,  are  sinusoidal,  and  if  there  is  no  power  lost  in  the 
reactance  coil  R2L2,  except  R2I2  (supplied  by  the  power  electro- 
motive force  BC  =  R2I),  OCA  will  be  a  right  triangle.  In  this 
case,  the  locus  of  C  will  be  a  semicircle  with  diameter  OA=E, 


126 


SINGLE-PHASE  CURRENTS. 


[Exp. 


and  the  locus  of  D  (the  current  locus)  will  be  a  semicircle*  with 
diameter  E-r-X,  at  right  angles  to  E.  The  theoretical  semi- 
circles are  shown  by  dotted  lines  in  Fig.  2. 

§  12.  If,  however,  the  power  consumption  in  the  reactance  coil 
is  more  than  7?2/*,the  locus  of  C  will  be  flattened  so  as  to  lie  inside 
of  a  semicircle.  This  would  be  the  case  in  a  reactance  with  iron, 
and  is  likely  to  be  the  case  even  in  a  coil  without  iron  on  account 
of  eddy  currents  in  the  copper.  Eddy  currents  have  the  effect  of 
increasing  R2  above  the  value  determined  by  direct  currents,  so 

that  RJ,  in  Fig.  2,  should  be 
increased  from  BC  to  BF. 
The  locus  of  C  is  affected 
by  energy  losses,  but  not  by 
wave  form. 

§  13.  The     current     locus 
(the   locus   of  D)    will   not 
be  affected  by  energy  losses, 
but  will  be  flattened  if  the 
current  is  not  a  sine  wave — 
due  to  an  impressed  electro- 
motive force  which  is  not  a 
distortion 
When 


o 
FIG.   3 


Resistance 
Values   of  7  and  0  for  different    sjne    wav€j    or 

values  of  R. 

caused  by  hysteresis. 

the  current  is  not  a  sine  wave,  the  apparent  value  of  X  varies 
somewhat  with  R ;  since  X  is  not  constant,  the  locus  of  D  is  not 
an  exact  semicircle. 

§  14.  Constant  Current  Operation. — It  is  seen  that  when  0  is 
large,  i.  e.,  when  X-+-R  is  large,  the  current  remains  nearly  con- 
stant, irrespective  of  any  variation  of  R  and  0;  between  Q  =  jo° 
and  0  =  90°  the  value  of  the  current  varies  only  6  per  cent.  This 
is  the  condition  for  constant  current  operation  and  is  obtained 

*  (§iia).  If  this  is  a  semi-circle,  I=(E-*-X)  sin  e  and  XI  =  E  sin  0- 
this  accords  with  facts  (see  Fig.  2,  Exp.  4-A)  and  the  proposition  is 
accordingly  proved. 


4-B]  CIRCLE  DIAGRAM.  127 

in  any  apparatus  by  means  of  high  reactance  within,  or  external 
to,  the  apparatus.  Constant  current  generators  (§8,  Exp.  3~A, 
§§27,  27a,  Exp.  3~B)  and  constant  current  transformers  (§§24, 
24a,  Exp.  5-C)  are  so  constructed. 

§  15.  Rectangular  Coordinates. — The  results  shown  in  polar 
coordinates,  in  Fig.  2,  should  also  be  shown  in  rectangular  coordi- 
nates, as  in  Fig.  3,  the  values  of  /  and  0  being  plotted  for  different 
values  of  the  total  resistance  of  the  circuit.  For  small  values  of 
R  it  will  be  seen  that  the  current  is  nearly  constant. 


CHAPTER  V. 
TRANSFORMERS. 

EXPERIMENT  $-A.  Preliminary  Study  and  Operation  of  a 
Transformer. 

PART  I.    INTRODUCTORY. 

§  I.  A  transformer  consists  of  three  elements :  a  core  of  lami- 
nated iron;  and  a  primary  and  a  secondary  winding  upon  this 
core.  The  two  windings  are  insulated  from  each  other  and 
usually  from  the  core;  they  are  in  close  proximity  to  each  other 
or  are  so  inter-spaced  that  practically  all  the  flux  which  passes 
through  one  must  pass  through  the  other — i.  e.,  there  is  the  least 
possible  magnetic  leakage. 

The  transformer  is  used  on  alternating  current  circuits  to 
increase  or  step-up  the  voltage,  or  to  decrease  or  step-down  the 
voltage,  in  the  ratio  of  the  number  of  primary  to  secondary  turns 
(S-L'.SZ);  there  is  a  corresponding  opposite  change  in  the  cur- 
rent in  the  ratio  S2 :  Slf  an  increase  in  voltage  being  accompanied 
by  a  decrease  in  current,  and  vice  versa.  It  is  chiefly  the  trans- 
former which  makes  alternating  current  superior  to  direct  cur- 
rent for  power  transmisssion,  for  it  makes  possible  a  high  poten- 
tial on  the  transmission  line,  with  consequent  copper  economy 
(§50,  Exp.  6-A),  and  any  desired  lower  potential  at  the  gen- 
erating and  at  the  receiving  apparatus. 

§  2.  In  operation,  the  primary  winding  is  connected  to  an  alter- 
nating current  supply  (see  Fig.  i).  A  current  flows  in  the 
primary  which  magnetizes  the  core,  i.  e.,  it  sets  up  an  alternating 
magnetic  flux  which  induces  an  electromotive  force  in  the  second- 
ary winding  and,  when  the  secondary  is  closed  through  a  resist- 

128 


5-A]  STUDY  AND  OPERATION.  129 

ance  or   other  load,   this   electromotive   force  causes   a   current 
to  flow. 

The  condition  very  nearly  attained  in  the  operation  of  a  trans- 
former is  the  transference  of  power  from  the  primary  to  the 
secondary  without  loss,  the  current  and  voltage  being  one  in- 


LOAD 


AAAAAA 


RESISTANCE 
FIG.  i.     Connections  for  loading  a  transformer. 

creased  and  the  other  decreased  in  the  ratio  of  turns.  Using 
subscripts  I  and  2  to  refer  to  the  primary  and  secondary,  respec- 
tively, the  product  E2I2  is  accordingly  nearly  equal  to  E-Jv 
being  in  fact  only  a  few  per  cent.  less. 

As  an  example,  let  Sl==  ioS2  in  a  20  K.W.  transformer.  The 
condition  very  nearly  attained  is 

Primary  watts  =20,000        Secondary  watts  ==20,000 

Primary  volts  =    1,000         Secondary  volts  =       100 

Primary  amperes  =.        20        Secondary  amperes          =      200 

On  account  of  losses,  however,  if  the  secondary  is  to  have  its 
full  rated  watts,  volts  and  amperes,  the  corresponding  primary 
quantities  must  be  slightly  more  than  the  amounts  shown  above. 

There  are,  accordingly,  the  following  losses :  lost  watts ;  lost 
volts;  lost  amperes. 

The  lost  watts  determine  the  efficiency  and  are  due  to  core 
losses  (hysteresis  and  eddy  currents)  and  copper  losses  in  both 
the  primary  and  secondary  windings. 

The  lost  volts  determine  the  regulation,  and  are  due  to  resist- 
ance drop  or  copper  drop   (which,  for  a  given  load,  is  propor- 
tional to  copper  loss,  see  §  30,  Exp.  5-B)  and  reactance  drop  due 
to  magnetic  leakage. 
10 


13°  TRANSFORMERS.  [Exp. 

The  lost  amperes  are  due  to  the  fact  that,  even  on  no  load, 
a  transformer  takes  a  certain  exciting  current  to  maintain  the 
flux  and  to  supply  the  core  losses.  For  a  more  detailed  discus- 
sion, see  Exps.  5~B  and  5~C. 

§  3.  Structurally,  transformers  are  of  two  types,  the  core  type 
— in  which  the  core  is  on  the  inside  and  not  the  outside  of  the 
coils;  and  the  shell  type — in  which  the  core  is  not  only  on  the 
inside  of  the  coils  but  also  encloses  them,  to  a  certain  extent,  on 
the  outside  so  as  to  form  a  divided  return  magnetic  circuit.  (See 
hand-books  and  text-books.)  Variations  in  structural  arrange- 
ments depend  on  commercial  considerations,  and  do  not  affect 
at  all  the  principle  of  operation. 

Transformer  losses  eventually  appear  as  heat  and  a  trans- 
former is  so  designed  that  this  heat  can  be  radiated  or  disposed 
of  without  exceeding  a  limiting  safe  rise  in  temperature.*  The 
magnetic  circuit  is  laminated  to  minimize  the  eddy  current  loss. 
For  all  usual  purposes,  the  magnetic  circuit  is  closed.  A  trans- 
former with  an  open  magnetic  circuit  takes  excessive  magnetizing 
current  and — while  it  might  be  used  for  some  special  purpose — 
it  is  never  used  for  power  and  lighting.  (The  "Hedgehog" 
transformer  was  of  this  type.) 

*(§3a)-  Heating  of  Transformers. — This  necessitates,  on  the  part  of 
the  designer,  a  consideration  of  radiating  surface,  etc.,  or  the  provision  of 
some  special  means  of  cooling.  The  radiating  surface  usually  found  nec- 
essary varies  between  2  and  4  sq.  in.  per  watt.  For  the  allowable  rise  of 
temperature,  see  Standardization  Rules  which  at  present  allow  a  rise  of 
50°  C.  above  the  air.  Run  at  higher  temperatures,  transformer  iron  ages, 
i.  e.,  the  core  losses  increase  in  the  course  of  time.  While  this  has  been 
true  of  the  iron  ordinarily  used  for  years  in  transformer  construction,  it 
is  less  true  of  the  improved  alloy  steels  which  are  being  introduced. 
Hence,  aging  ceases  to  be  a  factor  and  the  allowable  temperature  rise 
might  be  increased  as  much  as  the  insulating  material  will  stand.  Good 
insulation  will  stand  continuously  a  temperature  of  90°  C.  This  will 
increase  the  rating  of  a  given  size  transformer,  or  will  reduce  the  size 
and  cost  of  a  transformer  of  a  given  rating.  In  rating  new  iron  the 
allowable  magnetising  current,  and  not  temperature,  may  become  the  limit- 
ing consideration. 


5-A]  STUDY  AND  OPERATION.  I31 

§4.  In  the  majority  of  cases  transformers  are  used  on  con- 
stant potential  systems,  the  primary  and  the  secondary  potentials 
being  substantially  constant.  The  secondary  current,  accord- 
ingly, varies  with  the  load.  The  primary  current  varies  nearly 
in  proportion  to  the  secondary  current  and  to  the  load.  Trans- 
formers connected  in  different  parts  of  a  constant  potential 
system  are  in  parallel,  i.  e.,  the  primary  of  each  transformer  is 
connected  directly  across  the  line  so  as  to  receive  the  full  line 
voltage.  It  will  be  seen  (compare  Appendix  II.)  that  a  constant 
potential  transformer  is  essentially  a  constant  flux  transformer. 
Other  usesf  of  the  transformer  may  be  considered  special. 

Commonly,  transformers  are  made  for  single-phase  currents, 
there  being  a  single  primary  and  a  single  secondary  winding. 
On  polyphase  circuits,  several  single-phase  transformers  are  used 
(see  Exp.  6-A),  one  on  each  phase.  A  special  3-phase  trans- 
former, with  three  primary  and  three  secondary  coils  is  frequently 
used  (see  §26,  Exp.  6-A). 

§  5.  Object  and  Apparatus. — The  object  of  this  experiment  is 
to  familiarize  one  with  the  structure  and  general  behavior  of  a 
transformer  and  with  some  of  the  more  important  relations  be- 
tween the  different  quantities  involved  in  its  operation.  Sub- 
sequent experiments  go  more  fully  into  test  methods  (Exp.  5~B) 
and  theory  (Exp.  5~C),  parts  of  which  can  be  read  to  advantage 
in  connection  with  the  present  experiment. 

A  transformer  with  several  coils  having  the  same  number  of 
turns  is  well  suited  for  the  purposes  of  this  experiment.  The 

f  (§4a).  Here  may  be  mentioned  the  series  or  current  transformer,  with 
primary  in  series  with  the  line  and  secondary  supplying  current  for  an 
ammeter  or  wattmeter;  the  constant  current  transformer  for  supplying 
(constant  current)  arc  lights  from  a  constant  current  series  circuit,  one 
time  of  importance;  arc-light  transformers  for  supplying  (constant  cur- 
rent) arc  lights  from  a  constant  potential  primary  circuit,  depending  for 
their  operation  on  magnetic  leakage,  the  present  form  being  the  "  tub " 
type  with  movable  secondary  which  is  repelled  by  the  primary  and  counter- 
balanced by  weights. 


I32  TRANSFORMERS.  [Exp. 

following  outline  is  written  specifically  for  such  a  transformer 
having  four  equal  coils,  each  for  55  volts,  but  the  experiment 
may  be  modified  so  as  to  apply  to  a  transformer  wound  in  some 
other  manner.  See  Appendix  I.  for  polarity  and  ratio  tests  on 
a  commercial  transformer. 

PART  II.    TESTS. 

§  6  Polarity  Test ;  Series  and  Parallel  Connection  of  Coils. — 
Use  one  coil  as  a  primary  and  connect  it  (with  a  resistance  in 
series  as  a  safeguard)  to  a  55-volt  alternating  current  supply. 
Use  the  other  three  coils  as  a  secondary,  connecting  them  in  series 
in  such  a  manner  that  the  three  electromotive  forces  are  additive 
and  do  not  oppose  one  another.  To  prove  this,  measure  the 
electromotive  force  of  each  coil  and  the  electromotive  force 
across  the  three;  the  latter  value  should  equal  the  sum  of  the 
three  other  values.  This  establishes  the  polarity  of  the  coils. 

§7.  With  the  primary  circuit  unchanged,  join  the  three  sec- 
ondary coils  in  parallel.  In  doing  this  it  is  necessary  to  make 
sure  that  terminals  about  to  be  connected  together  are  of  like 
polarity,  as  in  connecting  batteries.  When  the  polarity  has  been 
already  established,  as  in  the  preceding  section,  the  proper  parallel 
connection  can  be  readily  made.  The  following  procedure,  how- 
ever, insures  the  right  connection  independent  of  previous  knowl- 
edge of  polarity,  two  coils  being  first  joined  in  parallel  and  the 
third  coil  being  then  joined  in  parallel  with  these  two.  To  deter- 
mine which  terminals  should  be  connected  together  to  connect 
two  coils  in  parallel,  join  a  terminal  of  one  coil  to  a  terminal 
of  the  other  coil  and  connect  a  voltmeter  to  the  two  remaining 
terminals.  If  the  voltmeter  reads  zero,  the  two  terminals  con- 
nected to  the  voltmeter  may  be  joined  together  and  the  two  coils 
will  be  in  parallel.  If  the  voltmeter  does  not  read  zero  or  very 
nearly  zero,  the  terminals  connected  to  the  voltmeter  cannot  be 
joined  together  without  causing  a  short  circuit  of  the  two  coils 


5-A1  .     STUDY  AND  OPERATION.  133 

which  would  give  rise  to  excessive  current  and  burn  out  the 
transformer. 

§  8.  Marking  Polarity.  —  It  is  convenient  and  common  to  desig- 
nate the  polarity  of  several  coils  by  some  systematic  marking; 
thus,  all  terminals  of  one  polarity  may  be  marked  prime  (')  and 
those  of  opposite  polarity  be  unmarked.  To  connect  coils  in 
parallel,  marked  terminals  are  connected  to  one  line  and  un- 
marked terminals  to  the  other;  to  connect  in  series,  the  marked 
terminal  of  one  coil  is  connected  to  the  unmarked  terminal  of 
the  next  coil,  as  in  Figs.  2,  3  and  5. 

In  marking  polarity  it  is  always  best  to  have  the  marked  sec- 
ondary terminals  of  the  same  polarity  as  the  marked  primary 
terminals.  Some  positive  convention*  of  this  kind  becomes  im- 
portant whenever  proper  polarity  is  essential,  as  in  the  case  of 
transformers  supplying  the  same  secondary  main,  transformers 
on  polyphase  circuits  (Exps.  6-A  and  7~A),  transformers  used 
for  reducing  the  current  or  voltage  supplied  to  wattmeters,  etc. 

For  polarity  and  ratio  tests  on  commercial  transformers,  see 
Appendix  I. 

§9.  Ratio  Test.f  —  Compute  and  verify,  experimentally,  the 
different  ratios  of  voltage  transformation,  E^-^E^  which  are 
possible  with  the  transformer.  At  any  instant  the  electromotive 
force  of  a  coil  is,  by  Faraday's  Law, 


- 

dt' 

where  5*  is  number  of  turns  and  <£  is  flux.  The  instantaneous 
value  of  the  voltage,  and  hence  the  effective  or  virtual  value, 
is  accordingly  proportional  to  the  number  of  turns,  and  the 
ratio  of  voltages  in  any  two  coils  is  the  ratio  of  the  number  of 
turns  in  the  coils.  (See  Appendix  I.) 

*  In  connecting  together  transformers  of  different  makes,  care  must  be 
taken,  for  their  polarities  may  be  indicated  by  different  systems. 

t  For  current  ratio  and  tests  on  commercial  transformers,  see  Appendix  I. 


134  TRANSFORMERS.  [Exp. 

If  any  combination  of  coils  gives  a  voltage  which  is  beyond 
the  range  of  the  voltmeter,  these  tests  can  be  made  by  using  a 
lower  supply  voltage;  it  may  be  found  convenient  to  connect 
the  high  potential  side  of  the  transformer  to  the  line,  thus  step- 
ping the  voltage  down  to  a  lower  voltage  in  the  secondary. 

§  10.  Prove  that  the  voltage  of  the  secondary  is  either  in  phase, 
or  180°  out  of  phase, f  with  the  primary  voltage.  To  do  this, 
join  together  one  terminal  of  the  primary  and  one  terminal  of 
the  secondary,  so  that  -the  two  windings  are  in  series ;  the  supply 
voltage  is  connected  to  the  terminals  of  the  primary.  Measure 
the  voltage  across  the  primary,  the  voltage  across  the  secondary, 
and  the  voltage  across  the  two,  measured  between  the  terminal 
of  the  primary  and  the -terminal  of  the  secondary  which  are  not 
joined  together.  Either  the  sum  or  the  difference  of  the  first 
two  readings  will  equal  the  third  reading;  whether  it  is  the  sum 
or  the  difference  will  depend  on  which  terminals  of  the  two 
windings  are  connected  together.  If  the  two  voltages  were  of 
different  phase,  the  total  would  be  found  to  be  not  the  algebraic 
sum  but  the  vector  sum. 

§11.  Use  as  an  Auto-transformer.  —  As  ordinarily  used,  a 
transformer  has  two  independent  circuits,  a  primary  and  a  sec- 
ondary, and  any  particular  winding  is  used  as  part  of  one  of 

f  (§  loa).  The  secondary  eletromotive  force  is  in  the  same  phase  as  the 
primary  counter  electromotive  force,  being  induced  by  (substantially)  the 
same  flux;  hence  it  is  opposite  in  phase  to  the  primary  impressed  or  line 
electromotive  force.  It  follows  that  the  secondary  current,  when  the 
transformer  is  loaded,  is  nearly  opposite  in  phase  to  the  primary  current, 
this  being  discussed  more  fully  in  Exp.  5-C  This  opposition  of  currents 
is  verified  by  the  auto-transformer  test,  §  n. 

That  primary  and  secondary  current's  are  opposite  to  each  other  in 
phase  may  be  further  illustrated  by  the  following  experiment.  Take  a 
straight  upright  core  surrounded  by  a  primary  circuit.  Place  around  it 
(loosely)  a  closed  ring  forming  a  secondary  circuit.  Connect  the  primary 
to  an  alternating  current  supply.  When  the  primary  circuit  is  closed,  the 
secondary  will  be  thrown  off  violently,  showing  that  the  currents  in  the 
two  circuits  are  in  opposite  directions.  The  secondary  ring  may  be  held 
down  by  threads,  so  as  to  float  as  a  halo. 


5-A] 


STUDY  AND  OPERATION. 


'35 


110  Volt  Supply 


Load 

Resistance 

FIG.  2.  Step-down  auto-trans- 
former, using  coils  A  B  C  D  as 
primary;  coil  D  is  also  used  as 
secondary. 


these  only.     In  the  auto-transformer,*  or  single-coil  transformer, 
part  of  the  windings  is  common  to  both  primary  and  secondary. 

Connect  the  transformer  coils  as  an  auto-transformer,  and 
verify  the  different  values  of  voltage  transformation.  To  do 
this  connect  all  coils  in  series  and  consider  any  one  or  more  of 
the  coils,  as  may  be  desired,  to  be 
primary  or  secondary.  Some  of 
the  coils  will  at  the  same  time 
form  part  of  both  primary  and 
secondary;  these  coils  will  carry, 
therefore,  both  the  primary  and 
the  secondary  currents,  which  are 
opposite  in  phase  (§  ica)  and  so 
give  a  resultant  current  approxi- 
mately equal  to  the  arithmetical 
difference  of  the  two. 

§12.  Connect  the  coils  as  a  step-down  auto-transformer  (Fig. 
2)  and  as  a  step-up  auto-transformer  or  "booster"  (Fig.  3). 
Using  suitable  resistances  as  a  load,  determine  the  current!  in 
each  coil,  in  the  resistance  and  in  the  supply  line  and  explain 
their  relative  values.  The  currents  and  voltages  for  other  com- 
binations of  coils  can  be  computed  and  compared,  or  determined 
experimentally.  Suppose  a  3 : 2  ratio  is  desired ;  with  A,  B,  C 
as  primary,  how  would  .the  use  of  C,  D  as  secondary  compare 
with  the  use  of  B,  C? 

§  13.  Advantages  of  the  Auto-transformer. — It  will  be  -found 
that  the  auto-transformer  requires  less  copper  than  a  transformer 
with  separate  primary  and  secondary  coils;  it  has,  therefore,  not 
only  lower  first  cost  but  less  copper  loss  and  copper  drop,  giving 
better  efficiency  and  regulation.  The  saving  in  space  on  account 

*  Also  called  "balance  coil"  or  "compensator";  the  term  auto-converter 
should  be  discarded. 

t  In  making  measurement  of  current,  it  will  be  found  convenient  to  use 
one  ammeter  and  a  3-way  ammeter  switch. 


[36 


TRANSFORMERS. 


[Exp. 


110  Volt  Supply 


A  B  C  as  primary  ;  coils  A  B  CD 
are  used  as  secondary. 


of  less  copper  makes  it  possible  to  reduce  also  the   iron  and 
iron  loss. 

This  advantage  of  an  auto-transformer  will  be  seen  to  be 
greater  the  nearer  the  ratio  of  transformation  is  I :  I.  For  a 
comparison  of  output  of  transformers  and  auto-transformers, 
see  §§8,  9,  Exp.  £-B.  An  auto-transformer  cannot  be  used 

when  complete  insulation  of  the 
primary  from  the  secondary  is 
necessary,  as  in  house  lighting  from 
high  potential  lines. 

The  step-down  auto-transformer 
of  Fig.  2  is  in  common  use  as  a 

starting  device   for   induction  mo- 
Load  Resistance 

FIG.  3.  Step-up  auto-trans-  tors>  givmg  a  lower  voltage  than 
former  or  booster,  using  coils  full  line  voltage  while  the  motor 

is  coming  up  to  speed;  see  Fig.  6, 
^^^  Exp.  7-A. 

A  common  use  of  the  step-up  arrangement  of  Fig.  3  is  as  a 
booster  to  raise  the  voltage  on  remote  parts  of  a  distribution 
system,  say  from  2,000  to  2,200  volts.  For  this  a  standard 
2,000/200  volt  transformer  can  be  used,  with  the  low-potential 
coil  in  series  with  the  primary  to  boost  the  voltage,  as  in  Fig.2, 
Exp.  7— B.  This  becomes  a  "  negative  booster  "  if  the  connec- 
tions of  the  low-potential  coil  (coil  D  in  Fig.  3)  are  reversed. 
(If  a  standard  transformer  is  to  be  tried  in  the  laboratory,  a 
loo-volt  circuit  may  be  boosted  to  no  volts,  or  reduced  to 
90  volts.) 

§  14.  Constant  Potential  Operation. — Transformers  are  usually 
operated  from  a  constant  potential  circuit,  so  as  to  transform — 
either  step-up  or  step-down — from  a  constant  primary  potential 
to  a  constant  secondary  potential. 

§  15.  Open  Circuit. — Connect  a  no-volt  alternating  current 
supply  circuit  across  two  of  the  transformer  coils  in  series,  as 


5-AJ  STUDY  AND  OPERATION.  *37 

a  primary.  Measure  the  no-load  primary  current,  70,  called  the 
exciting  current.  Predict,  and  then  measure,  the  value  of  /0 
when  the  two  primary  coils  are  in  parallel  and  connected  to  a 
55-volt  supply — i.  e.,  half  the  preceding  voltage.  Compare  the 
relative  values,  for  the  two  cases,  of  primary  turns,  ampere 
turns,  volts,  volts  per  turn  and  flux  density. 

Measure  /0  when  the  two  primary  coils  are  in  series,  and  con- 
nected to  a  55-volt  supply;  and  interpret  the  results  (see  Fig. 
2,  Exp.  5-B). 

Commercial  transformers  are  commonly  built  with  two  pri- 
maries for  connection  in  series  (for,  say,  2,200  volts)  or  parallel 
(for  i,  100  volts)  ;  and  two  secondaries  for  connection  in  series 
(for,  say,  220  volts)  or  parallel  (for  no  volts). 

§  16.  Operation  Under  Load* — Join  twof  of  the  coils  in  series 
to  form  a  primary  and  join  the  other  two  coils  in  series  to  form 
a  secondary — or  make  such  other  arrangement  of  coils  as  may 
be  desired.  Connect  the  primary  with  an  alternating  current 
supply — say  no  volts,  60  cycles — appropriate  to  the  arrange- 
ment of  coils  adopted.  A  voltmeter,  ammeter  and  wattmeter  are 
connected J  in  the  primary,  as  in  Fig.  i. 

§  17.  With  the  secondary  on  open  circuit,  measure  the  primary 
voltage,  the  primary  current  (in  this  case,  the  no-load  current, 
70)  and  the  primary  power  (in  this  case,  the  no-load  or  core 
losses,  W0). 

*  Time  should  not  be  spent  in  an  attempt  to  get  very  accurate  results 
in  this  test,  particularly  if  it  is  to  be  followed  by*  the  more  accurate  test 
by  the  method  of  losses,  Exp.  5-B. 

f  (§  i6a).  Where  there  is  a  choice  of  coils,  select  an  arrangement  which 
avoids  great  magnetic  leakage.  If  each  coil  forms  one  layer  or  section,  to 
take  the  first  two  for  primary  and  the  other  two  for  secondary  would  not 
be  a  good  arrangement.  In  a  commercial  transformer,  the  primary  and 
secondary  windings  are  so  placed  as  to  reduce  magnetic  leakage ;  to  secure 
this  end,  however,  all  the  windings  should  be  used,  that  is,  no  coil  should 
be  left  idle.  An  arrangement  of  coils  commonly  used  is  as  follows :  low, 
high,  high,  low,  potential. 

$  With  instruments  arranged  as  in  Fig.  i,  no  corrections  need  be  made. 
(See  Appendix  III.) 


13s  TRANSFORMERS.  [Exp. 

§  18.  Load  the  secondary  by  means  of  suitable  non-inductive 
resistance.  Change  this  resistance  by  steps  so  as  to  vary  the 
secondary  current  between  no  load  and  25  per  cent,  overload. 
At  each  step  measure  the  primary  voltage  Elt  current  Ilf  and 
power,  W i ;  also  the  secondary*  voltage  E2,  and  secondary  cur- 
rent I2.  The  product  of  the  secondary  voltage  and  current  will 
give  the  secondary  power  W2,  the  secondary  load  being  non- 
inductive.  In  practice,  a  load  of  incandescent  lamps  is  non- 
inductive,  but  not  so  a  motor  load. 

§  19.  Measure  the  resistance  of  primary  and  secondary.  (See 
§  15,  Exp.  5-B.) 

§20.  For  each  load,  compute  the  power  factor  (W^~-EJ^)\ 
also  the  angle  0  by  which  the  primary  current  lags  behind  the 
electromotive  force.  (Power  f actor  =  cos  6.) 

Plot  Jj,  W±,  power  factor,  6,  E2  and  W2  for  different  values 
of  I a,  as  in  Fig.  4.  Plot,  also,  the  copper  loss  for  primary 
(RJi2)  and  for  secondary  (R2I22)  and  the  core  loss  WQ  (the 
value  of  W±  on  open  circuit)  which  is  constant  at  all  loads,  as 
in  Fig.  8,  Exp.  5-B. 

Note  that  E2  decreases  with  load.  Determine  the  per  cent, 
regulation — the  per  cent,  increase  in  E2  in  going  from  full  load 
to  no  load. 

Note  the  current  ratio  for  different  loads.  It  will  be  seen  that 
as  the  transformer  becomes  loaded  (by  decreasing  resistance  in 
the  secondary)  the  secondary  current  becomes  more  nearly  equal 
to  the  primary  current  (multiplied  by  5i1-v-.S12).  In  a  loaded 
transformer,  primary  and  secondary  ampere-turns  are  practically 
(but  not  exactly)  equal. 

It  is  seen  that  in  a  transformer  there  is  a  loss  in  volts,  a  loss 
in  amperes  and  a  loss  in  watts,  this  last  determining  the  efficiency. 
While  best  for  illustrating  the  operation  of  a  transformer,  the 

*  By  means  of  suitable  transfer  switches  one  voltmeter  and  one  ammeter 
may  be  used  for  both  primary  and  secondary. 


5-A] 


STUDY  AND  OPERATION. 


139 


loading  method  is  not  so  good  for  the  accurate  determination  of 
efficiency  and  regulation.      These  can  be  computed  much  more 


100 


90 


70 


1500 


£1000 


500 


0  5  10  15  20  25 

SECONDARY  CURRENT;  AMPERES 

FIG.  4.      Curves  for  2,000/100  volt,  2  K.W.  transformer  ;  see  also  Fig.  8,  Exp.  s-B. 

accurately  from  the  losses,  determined  without  loading,  as  in 
Exp.  5-B. 

§21.  Load  the  transformer  with  an  inductive  load  and  take 
one  reading  of  the  instruments.  It  will  be  seen  that  the  sec- 
ondary voltage  is  somewhat  less  than  it  was  with  non-inductive 
load — that  is,  the  regulation  is  poorer.*  This  happens  when 
induction  motors  are  operated  from  transformers.  In  this  case 
the  secondary  current  is  lagging.  If  the  secondary  current  were 
leading,  the  secondary  voltage  in  some  cases  would  increase, 
instead  of  decrease,  with  the  load.  The  results  are  similar  to 
those  obtained  for  an  alternator ;  see  Exp.  3-6,  particularly  Fig.  7. 

§22.  Design  Data  and  Computation  of  Flux  Density. — Note 
the  construction  and  essential  dimensions  of  the  transformer, 

*  (§2ia).  If  the  leakage  reactance  of  a  transformer  is  small,  compared 
with  its  resistance,  the  regulation  may  be  better  at  low  than  at  high  power 
factor ;  compare  §  28,  Exp.  3-B. 


140  TRANSFORMERS.  [Exp. 

including  the  cross  section  of  the  magnetic  circuit  and  size  of 
wire,  but  do  not  remove  parts,  destroy  insulation  or  damage  the 
transformer  in  any  way  in  seeking  this  information.  Data  fur- 
nished by  the  maker  can  be  used  for  this  purpose. 

§  23.  Compute  the  current  density  in  amperes  per  square  inch 
and  in  circular  mils  per  ampere,  for  the  primary  and  the  sec- 
ondary windings.  Current  densities  from  1,000  to  2,000  circular 
mils  per  ampere  are  common,  but  less  copper  was  often  allowed 
in  early  transformers. 

§  24.  Compute  the  maximum  value  of  the  total  flux  in  C.G.S. 
lines  or  maxwells  (see  §9a,  Exp.  i-B)  ;  thus 


where  E  is  the  voltage  and  5  the  number  of  turns  for  any  coil, 
and  n  is  frequency.  The  quantity  E  -r-  S  is  the  volts  per  turn. 
For  proof  of  formulae,  see  Appendix  II. 

Compute  the  maximum  value  of  the  flux  density  in  gausses 
(flux  per  sq.  cm.)  ;  thus 

R  X  io8 
Flux 


where  A  is  the  cross  section*  of  the  core  in  sq.  cms.  If  A  is  in 
square  inches,  5max.  is  the  flux  density  in  lines  per  square  inch. 
If  A,  in  square  inches,  is  multiplied  by  6.45,  the  formula  gives 
£max.  in  gausses  —  for,  unfortunately,  this  mixture  of  C.G.S.  and 
English  units  is  in  common  use. 

§25.  The  computations  for  B  should  be  made  for  standard 
frequency  (60  cycles),  and  two  other  frequencies  (30  and  120) 
with  the  same  value  of  E.  If  values  of  A  and  5  are  not  obtain- 
able, assumed  values  may  be  assigned  for  practice  computations. 
If  the  cross  section  of  the  core  is  not  uniform,  B  will  have  dif- 

*  (§24a).  The  net  cross  section  is,  say,  15  per  cent,  less  than  the  gross 
cross  section  on  account  of  lamination. 


5-A]  STUDY  AND  OPERATION.  H1 

ferent  values  for  different  parts  of  the  magnetic  circuit.  From 
these  computations  it  can  be  seen  whether  B  will  be  more  or  less, 
if  a  transformer  is  operated  at  a  higher  or  lower  frequency  than 
rated  and  at  the  same  voltage.  (Note  in  what  manner  E  should 
be  changed  to  maintain  B  the  same  at  different  frequencies.) 
Practically,  transformers  are  run  at  different  frequencies  without 
changing  E,  if  the  frequency  is  not  too  far  below  the  frequency 
for  which  the  transformer  is  designed.  For  a  discussion  of  the 
effect  of  frequency  upon  core  loss,  see  §§  8-14,  Exp.  5~B. 

In  transformer  design,*  B  is  given  a  wide  range  (4,000-14,000 
gausses  at  60  cycles),  being  sometimes  greater  in  small  than  in 
large  transformers  and  greater  in  transformers  designed  for  low 
than  in  those  for  high  frequency.  In  design,  E  and  n  being 
given,  B  may  be  assumed  and  the  product  A  X  ^  determined. 
This  product  being  fixed,  the  designer  may  adjust  the  values  of 
A  and  5  to  suit  his  purpose,  increasing  A  and  decreasing  5  to  use 
more  iron  and  less  copper,  or  vice  versa. 

§  26.  From  the  formula  for  flux  density,  it  will  be  seen  that 
the  electromotive  force  of  any  coil  of  a  transformer  is  propor- 
tional to  the  number  of  turns  in  the  coil,  a  fact  already  noted. 
The  volts-per-turn  should  be  computed  as  a  constant  for  the 
transformer.  For  small  transformers  this  may  be  one  third  or 
one  half,  being  greater  for  large  transformers,  perhaps  2  to  4 
for  transformers  above  30  K.W.  The  reciprocal  gives  the  turns- 
per-volt.  The  volts-per-turn,  when  known  for  a  certain  type  and 
size  of  transformer,  may  be  used  as  a  design  constant. 

§27.  Other  data  of  interest  to  the  designer  (which  may  be  de- 
termined when  worth  while)  are  the  weight  of  copper  and  of  iron, 
total  and  per  K.W.  This  may  range  from  5  to  25  Ibs.  per  K.W. 
for  either  copper  or  iron.  The  space  factor  for  copper  is  the 

*  (§25a).  For  more  complete  design  data,  see  handbooks,  etc.  As  mag- 
netic material  is  improved,  higher  magnetic  densities  are  possible  for  the 
same  loss.  While  densities  of  4,000-8,000  were  used  with  ordinary  grades 
of  iron,  densities  of  6,000-12,000  are  now  common  with  alloy  steel. 


I42      .  TRANSFORMERS.  [Exp. 

ratio  of  the  cross  section  of  copper  to  the  total  cross  section  of 
the  windings,  i.  e.,  to  the  cross  section  of  copper  plus  insulation 
and  air  space.  Similarly  the  space  factor  for  the  iron  is  the 
ratio  of  its  net  to  gross  section. 


APPENDIX   I. 
POLARITY  AND  RATIO  OF  COMMERCIAL  TRANSFORMER. 

§28.  Polarity;  Alternating  Current  Method. — The  coils  are  con- 
nected in  series,  two  at  a  time,  and  notice  is  taken  whether  the 
voltage  around  the  two  is  the  sum  or  the  difference  of  the  separate 
voltages.  There  are  several  ways  in  which  this  can  be  carried  out. 

As  an  example,  let  us  take  a  transformer  with  two  primaries  for 
1,000  volts  each  and  two  secondaries  for  50  volts  each.     Connect  the 
two  i,ooo-volt  primaries  in  series  and  con- 
nect the  terminals  of  one*  of  the  primaries 
to   a   low   potential   supply   circuit,   say   50 
volts,  as  in  Fig.  5.     If  a  voltmeter  across 
the  two  coils  together  reads  zero,  reverse 
FIG.    5.      Polarity    test      the  connections  of  one  of  the  coils.     The 
by       alternating       current       yoltmeter  should  then  read  IQQ  vohs  acrQSS 
method. 

the  two  coils  together,  and  50  volts  across 

each  one  separately.  Terminals  A  and  B  are  now  of  one  polarity; 
terminals  A'  and  B'  are  of  the  opposite  polarity,  to  be  marked  with 
a  prime  (')  or  X- 

Each  secondary  is  then  connected  in  series  with  one  primary,  the 
primary  being  connected  to  the  5o-volt  supply  circuit;  the  secondary 
in  series  with  it  is  so  connected  that  the  voltmeter  reading  around  the 
two  coils  in  series  is  greater  (52.5  volts)  than  the  potential  from  the 
mains  (50  volts).  If  the  reading  is  less  (47.5  volts),  reverse  the 
secondary.  Secondary  terminals  are  marked  with  a  prime  (')  or  X 
to  correspond  with  the  primary. 

Small  transformers  are  commonly  so  wound  that,  when  the  primary 
and  secondary  leads  on  one  side  of  the  transformer  are  connected 

*  If  the  two  coils  in  series  were  connected  to  the  supply  circuit,  a  burn- 
out might  result  if  the  coils  were  opposed  to  each  other. 


-A]  STUDY  AND  OPERATION.  H3 

together,  the  voltage  measured  across  the  two  primary  and  secondary 
leads  on  the  opposite  side  will  be  the  sum  of  the  voltages  of  the  two 
windings. 

§  29.  Polarity;  Direct  Current  Method. — The  alternating  current 
method  is  usually  preferred,  but  sometimes  the  following  method  will 
be  found  convenient.  The  primary  is  supplied  with  a  direct  current 
sufficient  to  give  a  reading  on  a  direct  current  voltmeter  connected 
to  the  primary  terminals.  The  voltmeter  terminals  are  then  con- 
.  nected  to  what  are  supposed  to  be  corresponding  terminals  of  the 
secondary.  If,  when  the  primary  circuit  is  closed,*  the  voltmeter 
needle  is  thrown  in  the  same  direction  as  the  preceding  reading,  the 
voltmeter  has  been  connected  to  the  secondary  terminals  correspond^ 
ing  to  primary  terminals;  i.  e.,  the  voltmeter  lead  from  the  primary 
terminal  (')  or  X  is  connected  to  the  secondary  terminal  to  be  marked 
(')  or  X-  If  the  voltmeter  needle  is  thrown  in  the  opposite  direction, 
the  reverse  is  true. 

§  30.  Potential  Ratio. — Where  one  transformer  alone  is  to  be  tested, 
the  transformer  should  be  supplied  with  any  convenient  voltage  and 
the  voltage  of  each  circuit  measured  either  by  two  voltmeters,  one  of 
which  has  been  calibrated  in  terms  of  the  other,  or  by  one  voltmeter 
reading  direct  on  the  low  potential  side  and  with  a  multiplier  on  the 
high  potential  side.f 

When  one  transformer  has  been  tested  in  this  manner,  or  a  small 
potential  transformer  of  accurate  ratio  is  available,  two  transformers 
can  be  run  in  parallel  from  the  same  circuit  and  their  secondary 
voltages  on  open  circuit  compared  by  readings  taken  with  one  volt- 
meter or  with  two  voltmeters  whose  relative  calibration  is  known. 

If  the  secondaries  of  two  similar  transformers  are  connected  in 
series  and  in  opposition,  any  difference  will  be  shown  by  a  voltmeter 
connected  across  the  two. 

§  31.  Current  Ratio. — For  commercial  testing  of  the  ratio  of  a 
transformer,  test  may  be  made  by  comparison  of  primary  and 
secondary  currents  instead  of  voltages.  The  secondary  circuit  is 
short-circuited  through  an  ammeter  of  low  resistance  and  the 

*The  current  should  be  small  so  as  not  to  injure  the  voltmeter  by 
slamming  the  needle  when  the  circuit  is  made  and  broken. 

t  It  is  not  necessary  to  run  the  transformer  at  full  rated  potential. 
When  high  potentials  are  used,  due  caution  should  be  observed. 


M4  TRANSFORMERS.  [Exp. 

primary  and  secondary  currents  measured  when  a  proper  voltage  (a 
few  per  cent,  of  normal  primary  voltage)  is  applied  to  the  primary, 
so  that  about  the  normal  current  flows. 

§  32.  Circulating  Current  Test.— As  a  shop  test,  after  one  standard 
transformer  has  been  tested,  other  transformers  designed  for  the  same 
ratio  may  be  operated  from  the  same  primary  mains  and  tested  one 
at  a  time  by  connecting  each  secondary  to  be  tested  in  parallel  with 
the  secondary  of  the  standard,  terminals  of  the  same  polarity  being 
connected  together.  If  an  ammeter  shows  a  circulation  of  current 
through  the  secondaries,  the  two  transformers  are  not  of  the  same 
ratio. 

Commercially  a  small  difference  in  ratio  is  allowable  as  shown  by 
the  circulating  current,  which,  however,  should  never  exceed  one  per 
cent,  of  the  rated  full-load  current.  Instead  of  an  ammeter  a  suit- 
able fuse  may  be  conveniently  used,  and  more  safely  where  much 
difference  in  ratio  may  exist. 


APPENDIX    II. 
RELATION  BETWEEN  FLUX  AND  ELECTROMOTIVE  FORCE. 

§  33.  The  fundamental  relation  between  flux  and  electromotive 
force  is  expressed  by  Faraday's  law;  that  is,  in  a  closed  circuit*  of 
5  turns  embracing  a  varying  flux  <£,  the  induced  electromotive  force 
is  —  S-d(f>/dt.  In  a  transformer,  this  applies  alike  to  primary  or 
secondary.  In  the  case  of  a  primary  coil  this  induced  electromotive 
force  is  a  counter  electromotive  force  and  requires  to  overcome  it  an 
equal  and  opposite  impressedf  electromotive  force 


§  34.  Sine  Assumption.  —  Assuming  the  wave  of  electromotive  force 
to  be  a  sine  wave,  we  have 

e  =  £max.  sin  &t  ; 


*  Not  limited  to  a  transformer. 

t  (§33a).  The  actual  terminal  voltage  includes  also  resistance  drop,  thus 

The  resistance  drop,  however,  is  practically  negligible  in  the  primary  of  a 
transformer  on  open  circuit. 


S-A]  STUDY  AND  OPERATION.  H5 

=  £max.  sin  tot  dt ; 


The  maximum  value  of  the  flux  is 

<*>       =- 
and  the  flux  per  unit  area  is 


To  express  in  terms  of  effective  voltage,  substitute  \/2.E  for  Em&x. 
Multiplying  by  io8  to  change  from  C.G.S.  units  to  volts,  and  remem- 
bering that  w  is  27T  times  the  frequency  (w),  we  have  as  a  working 
formula  for  flux  density 


2*nSA 

It  follows  from  this  formula  that  a  constant  potential  transformer 
is  a  constant  flux  transformer.  It  also  follows  that,  if  a  certain  flux 
is  maintained  in  the  transformer,  the  voltage  in  any  coil  is  propor- 
tional to  the  number  of  turns  in  that  coil.  For  further  interpreta- 
tion, see  §§  24-26. 

§  35.  Without  Sine  Assumption. — We  have  the  fundamental  rela- 
tion 

d^  =  edt/S. 

Integrating  for  half  of  a  period  T,  during  which  time  the  flux  changes 
from  a  minus  to  a  plus  maximum, 

T 

X+<fr  i 

6  *S* 


Writing  i/n  for  T,  and  multiplying  by   io8  to  reduce  to  volts,  we 


This  is  true  of  any  shaped  electromotive  force  wave,  of  which  E&v  is 
ii 


146  TRANSFORMERS.  [Exp. 

the  average  value.  It  is  seen  that  the  maximum  value  of  the  flux 
depends  upon  the  average  and  not  the  effective  value  of  electromotive 
force.  If  we  let  the  form  factor  /  designate  the  ratio  of  effective  to 
average  value,  Eeff.  =/Eav.  and 


For  a  sine*  wave  f=i.i,  substituting  which  gives  the  formula  of  §  34. 

APPENDIX    III. 

USE  OF  A  WATTMETER,  VOLTMETER  AND  AMMETER;  ARRANGE- 
MENT OF  INSTRUMENTS  AND  CORRECTIONS  TO  BE  APPLIED. 

§  36.  The  ordinary  measuring  instruments  consume,  in  themselves, 
a  small  amount  of  power, — usually  only  a  few  watts.  In  many  cases 
this  can  be  neglected,  particularly  in  testing  apparatus  requiring 
considerable  power,  but  in  precise  measurements  of  small  quantities 
the  effect  of  these  losses  should  be  considered.  So  far  as  the  load 
run  in  the  present  experiment  is  concerned,  the  losses  in  instruments 
can  be  neglected;  the  errors  and  the  methods  of  correcting  for  them 
should,  however,  be  noted  for  use  whenever  necessary. 

Some  arrangements  of  instruments  introduce  larger  errors  than 
others.  Furthermore,  the  errors  can  be  readily  corrected  for  in  some 
cases  and  not  in  others. 

In  selecting  a  method  for  arranging  instruments,  choose  one  in 
which  the  errors  (even  if  large)  can  be  best  corrected  for,  or  else 
choose  one  in  which  the  errors  are  as  small  as  possible  and  no  cor- 
rection is  necessary.  So  far  as  convenience  is  concerned,  the  latter 
is  to  be  preferred. 

§  37.  The  Wattmeter. — A  wattmeter  has  two  coils :  a  series  or 
current  coil,  connected  in  series  with  one  line  of  the  circuit  as  an 
ammeter,  and  a  shunt  or  potential  coil,  connected  in  shunt,  from  one 
line  to  the  other,  as  a  voltmeter. 


*  (§3Sa).  For  a  sine  wave,  Eav  =  -£max  ;    and  Eeff  =  -^£max.-      See 

1/2 

page  37,  Bedell  and  Crehore's  Alternating  Currents.  Form  factor  is 
/  =  £eff  ^-£^v  =  i.i  for  a  sine  wave.  (Form  factor  was  first  used  by 
Roessler  as  Eav  -i-  £ef f ,  which  for  a  sine  wave  is  .9.) 


5-A] 


STUDY  AND  OPERATION. 


H7 


A  wattmeter*  may  be  connected  in  two  ways,  as  follows. 
§  38.  In  the  first  and  usual   method,  Fig.  6,  the  potential  coil  is 
connected  between  the  line  wires  on  the  supply  side.    In  this  method, 


Supply  Voltage  1 

—  • 

1! 

i-s 
11 

^^•HV^MX^ 

method 

_L 

J 

w 

1  1 

FIG.  6. 

First 

FIG.  7.     Second  method. 
Methods  for  connecting  a  wattmeter. 

the  wattmeter  reading  is  too  large,  including  not  only  the  true  watts 
of  the  load  but  also  the  RI2  loss  in  the  current  coil  of  the  wattmeter. 
This  error,  which  may  amount  to  several  watts,  can  frequently  be 
neglected;  correction  for  it  is  not  easily  made.  In  measuring  small 
amounts  of  power,  in  order  that  the  error  may  be  neglected,  the 
current  should  not  exceed  one  half  the  rating  of  the  current  coil  of 
the  wattmeter.  With  current  greater  than  this,  the  loss  in  the 
series  coil,  which  increases  with  the  square  of  the  current,  may  be 
too  large  to  neglect. 

§  39.  In  the  second  method,  shown  in  Fig.  7,  the  potential  coil  is 
connected  across  the  terminals  of  the  load.  In  this  method  the  watt- 
meter reading  is  also  too  large,  since  it  includes  the  RI2  loss  in  the 
potential  coil  of  the  wattmeter.  This  error  is  larger  than  the  error 
in  the  first  method  and  should  be  corrected  for  by  subtracting  E2  -+-  Rw 
from  the  wattmeter  reading,  E  being  the  line  voltage  and  Rw  the 

*  (§3?a).  Lag  Error. — A  wattmeter  measures  El  X  power  factor.  For 
accuracy,  a  wattmeter  must  have  the  resistance  of  the  potential  circuit  so 
large,  compared  with  its  reactance,  that  the  circuit  is  practically  non- 
inductive.  The  current  in  the  potential  circuit  is  then  practically  in  phase 
with  the  electromotive  force;  in  reality  it  lags  by  a  small  angle, 
0  =  tan'^Lw  -f-  R).  The  error  due  to  this  lag  angle  is  different  for  dif- 
ferent power  factors,  cos  0,  of  the  load.  The  true  watts  are  equal  to  the 
wattmeter  reading  multiplied  by 

COS0 

cos  Q  cos  (0  —  0) 

In  commercial  wattmeters,  at  commercial  frequencies,  this  correction  can 
be  neglected.  It  becomes  appreciable  on  higher  frequencies,  particularly 
on  loads  of  low  power  factor  and  at  low  voltages — i.  e.,  when  the  resistance 
of  the  potential  circuit  is  small. 


148  TRANSFORMERS.  [Exp. 

resistance  of  the  potential  coil  of  the  wattmeter.  This  correction 
might  be,  for  example,  5  watts  in  a  zoo-volt  wattmeter,  10  watts  in  a 
2oo-volt  wattmeter,  etc.  The  correction,  however,  is  exact  and  is 
readily  made,  the  value  of  R^  usually  being  given  with  the  instrument. 
In  precise  work,  this  method  should  be  used  and  the  correction  made. 
If,  however,  no  correction  is  to  be  made,  it  is  better  to  use  the  first 
method,  in  which  the  error  is  smaller. 

§  40.  Use  of  a  Voltmeter  with  a  Wattmeter. — When  a  voltmeter 
and  wattmeter  are  used  together,  the  voltmeter  should  (usually)  be 
connected  between  the  same  points  as  the  potential  coil  of  the  watt- 
meter. There  are,  therefore,  the  same  two  methods  of  connection  as 
with  a  wattmeter  alone. 

§  41.  First  Method. — The  voltmeter  is  connected  between  the  lines 
on  the  supply  side  of  the  wattmeter.  The  reading  of  the  voltmeter 
includes  the  RI  drop  in  the  current  coil  of  the  wattmeter ;  the  error  is 
small  and  may  often  be  neglected. 

§  42.  Second  Method. — The  voltmeter  is  connected  on  the  load  side 
of  the  wattmeter,  directly  to  the  terminals  of  the-  load.  The  volt- 
meter reading  is  now  correct.  The  wattmeter,  however,  includes  the 
watts  consumed  in  the  voltmeter.  The  reading  of  the  wattmeter 
should,  accordingly,  be  corrected  by  subtracting  E2  -f-  Rv,  where  Rv  is 
the  resistance  of  the  voltmeter.  The  whole  correction  for  the  watt- 
meter is  now  £2(i/Rw-{-  i/Rv),  which  is  to  be  subtracted  from  the 
wattmeter  reading. 

§  43.  Use  of  an  Ammeter  with  a  Wattmeter. — If  an  ammeter  is 
connected  in  circuit  on  the  load  side  of  a  wattmeter,  as  the  current 
coil  in  Fig.  6,  the  ammeter  reads  the  true  load  current.  The  watt- 
meter reading,  however,  includes  the  watts  loss  in  the  ammeter — a 
small  error  which  is  neglected. 

If  the  ammeter  is  connected  on  the  supply  side  of  the  wattmeter, 
no  error  is  introduced  in  the  wattmeter  reading ;  the  ammeter  reading, 
however,  is  too  large,*  since  it  includes  the  current  in  the  potential 

*  (§43a).  Ammeter  Correction. — The  ammeter  reading  can  be  corrected 
by  subtracting  (i/Rv-{-  i/R^W/L  The  current  which  flows  in  the  poten- 
tial circuits  of  the  voltmeter  and  wattmeter  is  E/Rv-{-E/R^.  This  is  in 
phase  with  the  electromotive  force  and  not  with  the  current,  and  must  be 
multiplied  by  the  power  factor  of  the  load  WIEI  to  get  its  component  in 
phase  with  the  current. 


5-A1  STUDY  AND  OPERATION.  H9 

coil  of  the  wattmeter.  This  error  can  be  neglected,  when  the  load 
current  is  large.  The  ammeter  reading  would  be  correct  if  the 
potential  coil  of  the  wattmeter  (and  voltmeter,  if  one  is  used)  were 
opened  when  the  ammeter  is  read;  sometimes  this  is  allowable  under 
steady  conditions,  but  simultaneous  readings  of  all  instruments  are 
usually  more  accurate. 

§  44.  Combinations  of  Instruments. — In  the  combined  use  of 
ammeter,  wattmeter  and  voltmeter,  the  best  method  to  use  depends 
somewhat  upon  the  conditions  of  the  test.  The  arrangement  of  Fig.  I 
is,  for  most  purposes,  as  good  as  any;  no  corrections  are  made. 
In  the  short-circuit  test  of  a  transformer,  the  reading  of  the  current 
is  most  important ;  hence,  Fig.  6,  Exp.  5-B,  the  ammeter  for  this  test 
can  best  be  connected  on  the  load  side  of  the  other  instruments.  For 
the  open-circuit  test,  voltage  is  important  and  not  current;  the 
ammeter  is,  therefore,  in  the  supply  line  and  the  instruments  arranged 
as  in  Fig.  I,  Exp.  5-B  (requiring  a  correction)  or  as  Fig.  I  of  this 
experiment  (requiring  no  correction  and  hence  simpler  to  use).  See 
§  3a,  Exp.  5-B. 

§  45.  Multipliers. — To  extend  the  range  of  a  voltmeter,  either  a 
series  resistance  (called  a  multiplier)  or  a  potential  transformer  can 
be  used.  The  potential  range  of  a  wattmeter  is  extended  in  the  same 
way. 

To  extend  the  range  of  an  ammeter,  a  current  transformer  is  used; 
the  primary  of  the  transformer  is  connected  in  series  with  the  line, 
the  secondary  being  short-circuited  through  the  ammeter.  The  cur- 
rent range  of  a  wattmeter  is  extended  in  the  same  way. 

The  ratio  of  transformation  of  any  potential  or  current  trans- 
former must  be  accurately  known,  and,  for  a  current  transformer, 
this  ratio  must  be  known  in  connection  with  the  particular  instru- 
ment and  secondary  leads  with  which  it  is  to  be  used.  Any  small 
phase  shifting,  due  to  the  fact  that  primary  and  secondary  quantities 
are  not  exactly  in  phase  opposition,  introduces  no  error  in  the  use 
of  instrument  transformers  with  ammeters  or  voltmeters,  but  with 
wattmeters  such  phase  shifting  may  introduce  considerable  error  and 
needs  to  be  taken  into  consideration  for  accurate  work.  For  a  com- 
plete discussion,  see  Electric  Measurements  on  Circuits  Requiring 
Current  and  Potential  Transformers,  a  paper  by  L.  T.  Robinson,  read 
at  the  June,  1909,  meeting  of  the  A.  I.  E.  E. 


15°  TRANSFORMERS.  [Exp. 

EXPERIMENT  5-B,  Transformer  Test  by  the  Method  of 
Losses. 

§  i.  Introductory. — The  losses  in  a  transformer  are  the  core 
loss,  which  is  dependent  upon  and  varies  with  voltage,  and  the 
copper  (and  load)  losses,  which  are  dependent  upon  and  vary 
with  current.  The  most  accurate*  and  the  most  convenient 
method  for  testing  a  transformer  is  to  measure  these  losses  sepa- 
rately, without  loading  the  transformer,  and  compute  the  effi- 
ciency and  regulation. 

This  requires  two  simple  tests,  each  employing^  a  voltmeter, 
ammeter  and  wattmeter:  an  open-circuit  or  no-load  test  for 
determining  the  no-load  or  core  loss  and  the  exciting  current 
at  various  voltages,  particularly  at  normal  voltage;  and  a  short- 
circuit  test  at  a  low  voltage  (a  few  per  cent,  of  normal)  for 
determining  the  copper  and  load  losses  and  impedance  drop  for 
various  currents,  particularly  for  normal  full-load  current.  The 
latter  test  gives,  also,  the  equivalent  resistance  and  leakage  react- 
ance of  the  transformer. 

Measurements  are  also  made  of  primary  and  secondary  re- 
sistance. ...  • 

§2.  This  method  may  be  employed  in  testing  any  transformer, 
whether  it  is  intended  for  constant  potential,  .constant  current  or 
other  service ;  the  method  will  be  described  in  detail  with  refer- 
ence to  its  application  to  a  constant  potential  transformer. 

*(§ia).  This  is  most  accurate  for  the  reasons  explained  in  §  ib,  Exp. 
2-B.  It  is  not  practicable  to  determine  efficiencies  accurately  by  loading 
a  transformer  (§  16,  Exp.  5-A)  and  measuring  the  input  and  output 
directly — unless  exceeding  care  be  taken — the  two  quantities  measured 
being  so  nearly  equal.  The  indirect  method  of  losses  is,  furthermore, 
most  convenient  because  no  load  is  required  and  no  high-potential  meas- 
urements are  necessary. 

f  In  many  cases  the  same  instruments  can  be  used  in  the  two  tests ;  .com- 
pare §44.  Two  similar  tests  are  made  in  testing  alternators;  see  §9, 
Exp.  3-B. 


5-B] 


TEST  BY  LOSSES. 


In  a  constant  potential  transformer  the  magnetization  and 
hence  the  core  loss  and  exciting  current  are  (substantially)  the 
same  at  all  loads,  being  dependent  upon  voltage  and  not  upon 
current.  The  copper  and  load  losses,  on  the  other  hand,  depend 
upon  current  and  vary  with  the  load.  (In  a  constant  current 
transformer,  the  conditions  are  reversed;  copper  losses  are  con- 
stant and  core  loss  varies  with  the  load.) 


PART  I.    OPEN-CIRCUIT  TEST. 

§  3.  With  the  secondary  open,  measurements  are  made  on  the 
primary  with  ammeter,  voltmeter  and  wattmeter,  one  method 
for  making  the  connec- 
tions* being  shown  in 
Fig,  i.  Although  any 
coil  or  combination  of 
coils  could  be  used  as  a 
primary  in  this  test,  it 
is  most  convenient  to 
use  a  low  potential  coil 
(50,  100  or  200  volts) 


Jl 


FIG.  i.  One  method  of  connection  for  open- 
circuit  test  for  core  loss  and  exciting  current. 
See  §  33  for  method  of  connecting  instru- 
ments requiring  no  corrections. 


as  primary  to  suit  the  instruments  and  supply  voltage  available; 
furthermore,  there  is  less  danger  in  working  on  the  low  potential 
side. 

For  the  same  degree  of  magnetization,  the  exciting  current 
in  a  loo-volt  coil  is  ten  times  as  large  as  m  a  i,ooo-volt  coil,  the 
ampere  turns  being  the  same.  It  is,  accordingly,  a  simple  matter 

*(§3a).  For  selection  of  instruments,  see  §44.  The  arrangement  of 
instruments  shown  in  Fig.  i  should  be  followed  when  the  highest  accuracy 
is  desired;  the  wattmeter  reading  is  to  be  corrected  by  subtracting 
E2(i/RVf-}-  iARv),  which  is  the  power  consumed  in  the  potential  coils  of 
the  wattmeter  and  voltmeter. 

It  is,  however,  much  simpler — and  in  many  cases  sufficiently  accurate — to 
arrange  the  instruments  as  in  Fig.  i  of  Exp.  5-A,  and  to  make  no  correc- 
tion. See  Appendix  III.,  Exp.  5~A. 


152  TRANSFORMERS.  fExp. 

to  reduce  the  exciting  current  measured  on  a  coil  of  one  voltage 
to  its  value  for  a  coil  of  another  voltage.  The  watts  core  loss 
is  the  same  measured  on  one  coil  as  on  another,  for  the  same 
magnetization. 

CAUTION.  If  two  coils  are  to  be  connected  in  parallel  or  series, 
to  avoid  a  burn-out  it  is  necessary  to  first  make  sure  of  their  polarity, 
as  described  in  §§  6  and  28,  Exp.  5~A. 

Be  careful  of  the  high-potential  terminals  in  this  test.  It  should  be 
made  impossible  for  loose  wires,  or  for  persons  making  measure- 
ments, to  come  in  contact  with  these  terminals.  Although  the  testing 
current  and  instrument  are  all  of  low  voltage  and  although  the  high- 
potential  coil  is  open  and  has  no  current  in  it,  the  potential  is  there 
and  must  be  respected. 

§  4.  Data. — At  normal  frequency,  say  60  cycles,  vary  the  volt- 
age (§45)  from  say  J  to  ij  normal  and  determine  the  core 
loss  WQ,  and  exciting  current  70  for  various  voltages.  Note  the 
frequency  at  which  the  test  is  made.  It  is  desirable  that  the 
frequency  be  maintained  constant,  and  that  the  voltage  be  of 
sine  wave-form. 

Take  very  accurate  readings  at  two  points  (within,  say,  5  or 
10  per  cent,  of  half  and  full  voltage)  by  taking  at  each  of  these 
points  a  series  of  five  readings  and  averaging.  This  two-voltage 
method  is  very  convenient,  since  normal  and  half  voltage  (as 
55/1 10  or  110/220)  are  often  available — or  their  equivalent  can 
be  obtained  by  series  and  parallel  connections,  as  described  in 
§  46.  As  will  be  seen  later,  Figs.  2  and  5,  it  is  very  accurate  for 
transformers  built  of  ordinary  iron,  at  normal  and  higher  fre- 
quencies, but  not  at  frequencies  far  below  normal.  For  trans- 
formers with  improved  iron,  the  two-voltage  method  is  not  cor- 
rect unless  one  observed  point  is  taken  very  near  full  voltage, 
little  error  being  then  introduced  by  obtaining  values  for  full 
voltage  from  the  curves. 

§  5.  If  possible,  repeat  the  data  at  a  second   frequency.      If 


S-B] 


TEST  BY  LOSSES. 


'53 


the  frequency  is  higher  than  normal,  complete  data  can  be  taken 
as  before.  If  the  frequency  is  lower  than  that  for  which  the 
transformer  is  intended,  the  core  loss  and  exciting  current  will 
be  greater  and  the  voltage  should  not  be  raised  so  that  they 
become  excessive  for  the  transformer  or  the  instruments;*  WQ 
should  not  exceed  say  twice  and  /0  four  or  five  times  their 
respective  values  at  normal  frequency.  It  will  be  understood, 
however,  that  these  limits  are  only  arbitrary. 

§  6.  Curves  for  Exciting  Current. — For  each  frequency,  plot 
a  curve  showing  the  exciting  current  for  different  voltages,  as 
in  Fig.  2.  Locate  by  heavy  black  dots  the  two  points  accurately 


0.605  Amp.=  3.03  percent. 
IH=0.416 
IM=  0.439 


I0=0.430  Amp.  =2.15  percent. 
IH=0.320    "      =1.60   "       " 


20 


40 


100 


130 


140 


VOLTS 


FIG.  2.  Observed  exciting  current  for  varying  voltage  at  different  frequen- 
cies;  2  K.W.  transformer,  loo-volt  coil.  To  obtain  the  exciting  current  for 
the  2,ooo-volt  coil,  divide  these  values  by  20. 

*  (§5a).  The  current  coil  of  the  wattmeter  has  a  certain  rated  current 
carrying  capacity  which  should  not  be  exceeded  for  any  length  of  time. 
It  may,  however,  be  exceeded  for  a  few  moments  only  by  30  or  even  40 
per  cent. ;  readings  are  taken  quickly  and  the  wattmeter  is  then  cut  out. 


154  TRANSFORMERS.  [Exp. 

determined  at  about  half  and  full  voltage  and  note  how  closely 
a  straight  line  drawn  through  them  coincides  with  the  curve 
through  the  working  range.  (It  will  be  found  that  this  straight 
line  construction,  based  on  two  readings,  can  not  be  used  when 
the  transformer  is  worked  at  a  very  high  flux  density,  as  in  the 
29-cycle  curve  of  Fig.  2.) 

§  7.  Take  from  each  curve  the  value  of  /0  for  normal  voltage 
(§48).  Resolve  the  exciting  current,  70,  into  two  components: 
the  in-phase  power  component  /H  (which  supplies  the  core  losses 
due  to  hysteresis  and  eddy  currents)  ;  and  the  quadrature  mag- 
netizing* component,  /M.  These  are  determined  by  the  following 
relations  : 


The  value  of  W  '0  is  taken  from  curves,  Fig.  3  and  Fig.  5, 
described  in  the  next  paragraph.  At  no  load,  power  factor 
=  IH~!O. 

If  the  transformer  under  test  had  a  core  made  of  improved 
steel,  with  less  core  loss,  the  component  /H  would  be  somewhat 
less  than  indicated  in  Fig.  2.  On  account,  however,  of  the  very 
much  greater  value  of  the  component  /M  (due  to  the  higher  flux 
density  common  with  such  iron),  the  total  exciting  current  70 
would  be  greater  than  shown  in  Fig.  2.  Furthermore,  the  point 
for  normal  voltage  would  be  near  the  knee  of  the  curve,  so  that 
the  straight  line  construction  would  not  be  accurate,  as  has 
already  been  pointed  out. 

Compute,  as  in  Fig.  2,  the  values  of  70,  IK  and  IM  as  per  cent. 
of  the  normal  full-load  current  of  the  coil  on  which  the  test 
is  made;  thus,  the  full-load  current  for  a  loo-volt  coil  of  a 
2  K.W.  transformer,  Fig.  2,  is  20  amperes.  Expressed  as  per 
cent.,  the  results  will  apply  to  any  coil. 

*  Usage  is  not  fixed  in  regard  to  the  terms  "  exciting  "  and  "  magnet- 
izing "  currents,  these  terms  being  not  infrequently  interchanged. 


S-B] 


TEST  BY  LOSSES. 


FIG.  3.  Watts  core  loss  for  varying 
voltage  at  different  frequencies ;  2  K.W. 
transformer,  loo-volt  coil. 


§  8.  Curves  for  Core  Loss. — The  readings  of  the  wattmeter  in 
the  open-circuit  test  (after  corrections  are  made,  if  there  are 
any,  §  3a)  gives  the  core  loss 
plus  a  small  RIQ2  loss  due  to 
the  heating  effect  of  the  ex- 
citing current.  The  latter  loss 
can  be  computed  and  deducted 
from  the  wattmeter  reading; 
it  will  generally  be  found 
negligible.* 

The  curves  showing  the 
change  of  core  loss  with  volt- 
age can  be  plotted  on  ordinary 
cross-section  paper  as  in  Fig. 
3 ;  a  derived  curve,  showing 
the  variation  of  core  loss  with  frequency,  being  plotted  as  in 
Fig.  4.  It  is  much  better,  however,  to  use  a  logarithmic  scale 
for  ordinates  and  abscissae,  in  which  case  the  curves  become 
(within  limits)  practically  straight  lines.  For  this  purpose,  it 
is  convenient  to  use  logarithmic  cross-section  paper.f 

Above  normal  voltage,  as  higher  densities  are  reached,  the 
curve  tends  to  bend  upwards,  due  to  the  fact  that  the  hysteretic 
exponent  (which  has  a  value  of  about  1.6  up  to  10,000  gausses) 
becomes  greater.  Transformers  with  improved  iron  are  run 
at  higher  densities,  so  that  at  normal  frequency  this  bend  may  be 
reached  at  normal,  or  even  below  normal,  voltage. 

§  9.  For   each   frequency  plot  a   curve  on   logarithmic  paper 

*  (§  8a.)  Although  any  Rio2  loss  should  be  deducted  for  obtaining  true 
iron  loss,  for  the  calculation  of  efficiency  it  is  better  not  to  make  such  a 
deduction  but  to  include  the  Rio2  loss  with  the  iron  loss  Wo. 

f  This  paper  can  be  obtained  from  the  Cornell  Cooperative  Society,  or 
Andrus  -and  Church,  Ithaca,  N.  Y. 

The  same  results  can  be  obtained  on  plain  paper  by  plotting  the  loga- 
rithms of  the  observed  quantities — a  laborious  process — or  by  using  a 
slide  rule  as  a  scale. 


60 


40 


cc 
2  20 


9-2 


I56  TRANSFORMERS.  [Exp. 

showing  the  core  loss,  W0,  for  different  voltages,  as  in  Fig.  5. 
Locate  by  heavy  black  dots  the  two  points  accurately  deter- 
mined at  about  half  and  full  voltage.  Draw  a  straight  line 
through  these  points  and  note  that  at  normal  and  higher  fre- 
quencies this  straight  line  gives  the  curve  accurately  through 

the  working  range,   so  that 
WQ  for  normal  voltage  can 

KAL_  'V 

be  readily  obtained  from  it. 
At  frequencies  much  below 
f.32  normal,  and  at  high  flux- 
densities,  this  straight  line 
relation  may  not  hold. 

10r*  If  it  was  impossible  to  get 

data    for   any   curve    up   to 

0         20       40       60        80      100      120      140  1          i/  j     ^ 

FREQUENCY:  CYCLES  PER  SECOND  normal   valtage,   extend   the 

FIG.  4.    Watts  core  loss  for  different  curve  that   far  as  a  dotted 

frequencies   at   normal  voltage;   2   K.   W.  ^  ^      extension      is 

transformer. 

quite  accurate  at  frequencies 

n^ar  normal  or  higher,  but  can  not  be  depended  upon  at  fre- 
quencies way  below  normal.  The  curves,  however,  can  be  more 
readily  and  more  accurately  extended  on  logarithmic  paper  than 
on  ordinary  coordinate  paper. 

§  10.  The  slope  of  these  curves  (the  actual  tangent  with  the 
horizontal)  is  the  exponent  (a)  of  E  or  B  in  the  formula  show- 
ing the  law  of  core-loss  variation  for  different  voltages  and 
flux  densities  at  a  constant  frequency; 

W  oc  Ea  oc  B\ 

This  exponent    (a)    should  be  determined  and  interpreted;  see 

§§49-51- 

§11.  Variation  of  Core  Loss  with  Frequency. — On  the  same 
logarithmic  sheet,  see  Fig.  5,  plot  a  'derived  curve  showing  the 
core  loss,  WQ,  at  normal  voltage  for  different  frequencies.  If 


5-B] 


TEST  BY  LOSSES. 

Frequency;  Cycles  per  Second 


100       120     140     160  180  200 


100       120     140    160  180  200 
VOLTS  (Logarithmic  Scale) 

FIG.    5.      Curves    for    core  loss  plotted  with   logarithmic  scale. 

there  are  data  for  only  two  frequencies  and  hence  only  two 
points  on  this  curve,  plot  it  as  a  straight  line — as  it  will  be 
practically  a  straight  line  through  a  considerable  range.  (When 
the  test  is  made  at  only  one  frequency,  see  §  52.) 

The  slope  of  this  line  gives  the  exponent   (b),  showing  the 
law  of  core-loss  variation  for  different  frequencies  at  constant 

voltage :  TJ7          6 

W0  oc  w6. 

The  exponent  (b)  should  be  determined;  see  §51.  Plotted  on 
ordinary  coordinate  paper,  this  curve  appears  as  in  Fig.  4. 


i58  TRANSFORMERS.  [Exp. 

§  12.  Fig.  5  also  shows  a  core-loss  curve  corrected  for  60 
cycles,  although  no  measurements  were  made  at  that  frequency. 
Such  a  derived  curve  can  be  drawn  for  any  desired  frequency. 

§  13.  It  is  seen  that  core  loss  becomes  less  as  frequency  is 
increased.  A  transformer  designed  for  one  frequency  can, 
therefore,  be  operated  more  efficiently  at  a  higher  frequency,  the 
voltage  remaining  unchanged.  Operated  at  a.  lower  frequency, 
however,  the  transformer  will  have  larger  core  loss  and  will 
therefore,  heat  up  more — unless  operated  at  a  lower  voltage 
and  reduced  output. 

The. transformer  to  which  Fig.  5  refers  is  seen  to  have  the 
same  core  loss  (41.6  watts),  and  so  would  have  the  same  tem- 
perature rise,  when  run  at  81  volts,  29  cycles;  98  volts,  60  cycles; 
100  volts,  66.2  cycles;  118  volts,  140  cycles.  The  volt-ampere 
capacity  of  any  transformer  is,  accordingly,  less  at  lower  fre- 
quencies. For  the  same  capacity,  a  larger  and  more  expensive 
transformer  is  required. 

If  transformers  were  the  only  consideration,  the  frequencies 
of  125  and  133  cycles  in  early  use  would  not  have  been  abandoned 
for  lower  ones.  (See  §3,  Exp.  3~A.) 

§  14.  Flux  Densities. — Compare  the  flux  densities  at  different 
frequencies,  for  normal  voltage.  The  values  of  flux  density 
can  be  computed,  as  in  §§  33-35,  Exp.  5-A,  if  the  number  of 
turns  and  iron  cross-section  are  known.  Without  calculating 
the  actual  values  and  without  knowing  the  construction  data  of 
the  transformer,  the  relative  values  can  be  found  by  the  rela- 
tion B  oc  E^-n.  Thus,  if  at  60  cycles  B  is  taken  as  i.o,  the 
flux  density  at  30  cycles  is  2.0,  at  120  cycles  0.5,  etc. 


OF    THE 

UNIVERSITY 

OF 


s-B]  -"TEST  BY  LOSSES.  *59 

PART  II.    RESISTANCE  MEASUREMENTS. 

§15.  Data.  —  The  primary  and  secondary  resistances  are  meas- 
ured by  means  of  a  bridge  or  by  direct  current,  fall-of-potential 
method  (§  17,  Exp.  i-A).  Disconnect  the  voltmeter  before  the 
current  is  thrown  off,  to  avoid  damage  by  jnductive  kick.  Avoid 
heating  the  coils  and  so  causing  their  resistance  to  increase;  the 
testing  current  should  not  exceed  25  per  cent,  of  the  full-load 
current  for  the  coil,  or  should  not  be  long  continued.  The  range 
of  the  ammeter  to  be  used  is  thus  determined  from  the  known 
value  of  full-load  current. 

The  range  of  voltmeter  is  found  by  assuming  an  approxi- 
mate value  for  resistance  drop;  thus,  if  the  resistance  drop 
were  one  per  cent,  in  primary  or  secondary  for  full-load  current, 
this  would  be  10  volts  in  a  i,ooo-volt  coil  and  I  volt  in  a  100- 
volt  coil.  If  only  one  fourth  of  full-load  current  were  used  toi 
testing,  the  voltage  readings  would  be  2.5  and  0.25  volts,  re- 
spectively. 

The  resistance  measurements  by  direct  current  are  to  be 
used  as  a  check  and  for  comparison  with  the  results  obtained 
in  the  short-circuit  test. 

Temperature  conditions   should  be   taken  account  of    (§22). 

§  1  6.  Equivalent  Primary  Resistance.  —  From  the  measured 
values  of  R^  and  R2,  compute  the  equivalent  resistance*  R, 

*  (§  i6a).  The  equivalent  resistance  R  must  have  such  a  value  that 


Dividing  by  I?  and  writing  the  ratio  of  turns  (Si  -4-  S2)  in  place  of  the  ratio 
of  currents  (72-=-/i),  we  have 


It  is  obvious,  also,  that  R—  (copper  loss)  -f-  If. 

Any  resistance  in  the  secondary,  either  within  the  transformer  or  in  the 
external  circuit,  has  the  same  effect  as  though  it  were  multiplied  by  the 
square  of  the  ratio  of  turns  and  placed  in  the  primary  circuit.  It  may 
be  noted  here  that  the  same  is  true  of  reactance. 


160  TRANSFORMERS.  [Exp. 

which  is  the  joint  resistance  of  the  primary  and  secondary  in 
terms  of  the  primary: 


The  value  thus  determined  will  be  used  for  comparison  with  the 
value  (R  =  Wc-t-I2)  determined  from  the  copper  loss  in  the 
short-circuit  test. 

PART  III.    SHORT-CIRCUIT  TEST. 

§  17.  Method   of   Test. — For   the   short-circuit   test,   the   sec- 
ondary (low-potential  side)  of  the  transformer  is  short  circuited 

and    the    primary     ( high- 


Is -o    potential  side)   is  supplied 

o        «•- 

o 
o 
o 


"i  o    with  a  small  difference  of 

*"« 

Q-t;    potential,  just  sufficient  to 

.3  "    cause  the  full-load  or  de- 

FIG.    6.     Connections    for    short-circuit    test    sired  Current  to  flow.     This 
for  copper  loss  and  impedance  voltage. 

is  rarely  more  than  5  or  o 

per  cent,  of  normal  voltage.  Instruments*  are  connected  in  the 
primary,  as  shown  in  Fig.  6.  The  current  might  be  measured 
by  an  ammeter  in  either  circuit,  but  it  is  betterf  to  have  the 
ammeter  on  the  primary  side  with  the  voltmeter  and  wattmeter. 

*  (§  I7a).  The  most  important  reading  to  have  correct  is  that  of  the 
ammeter,  since  the  wattmeter  reading  varies  as  /2  and  the  voltmeter  read- 
ing varies  as  /,  and  all  results  are  calculated  for  values  of  current.  For 
this  reason  it  is  well  to  place  the  ammeter  directly  in  the  primary  circuit, 
as  in  Fig.  6,  in  which  case  no  correction  is  necessary.  If  the  ammeter  is 
connected  in  the  supply  line,  as  in  Fig.  I,  and  the  instruments  are  read 
simultaneously,  a  small  error  is  introduced  (tending,  in  this  case,  to  favor 
the  transformer)  unless  a  correction  is  applied.  See  Appendix  III.,  Exp. 
5-A.  Connected  as  in  Fig.  6,  the  wattmeter  usually  needs  no  correction ; 
but  for  the  accurate  measurement  of  small  power  the  method  of  con- 
nection shown  in  Fig.  7,  Exp.  5-A,  should  be  used  and  a  correction 
applied. 

For  selection  of  instruments,  see  §44. 

t  (§i7b).  It  is  important  for  accuracy  to  have  the  short  circuit  of  the 
secondary  as  "  short "  as  possible,  i.  e.,  with  practically  zero  impedance 


, 


]  TEST  BY  LOSSES.  lj 

In  this  test,  when  full-load  current  flows  in  the  primary,  full- 
load  current  flows  in  the  secondary;  when  half-load  current 
flows  in  the  primary,  half-load  current  flows  in  the  secondary, 
etc.  The  flux  density  is  very  low,  so  that  there  is  practically 
no  core  loss.  The  wattmeter  reading  gives,  therefore,  the  total 
copper  losses — both  primary  and  secondary — for  any  particular 
current.  Included  with  the  copper  losses  are  the  load  losses. 

§  1 8  Load  Losses. — Load  losses  are  due  chiefly  to  eddy  cur- 
rents in  the  copper  and  are  greatest,  therefore,  in  large  solid 
conductors.  They  have  the  effect  of  causing  a  greater  loss  in  a 
conductor  when  traversed  by  alternating  current  than  when 
traversed  by  direct  current,  the  resistance  being  apparently  in- 
creased. The  term  load  losses  includes  all  losses*  which  in- 
crease with  load  and  depend  upon  current,  over  and  above  the 
copper  losses  as  determined  by  direct  current.  Evidently  such 
losses  should  be  taken  into  consideration  in  calculating  efficiency,! 
and  the  Standardization  Rules  of  the  A.  I.  E.  E.  so  specify. 

outside  of  the  transformer.  An  ammeter  and  its  leads  in  the  secondary, 
sometimes  used,  tends  to  give  the  transformer  a  poorer  regulation  and 
efficiency.  It  is  instructive,  however,  before  taking  readings,  to  insert  an 
ammeter  in  the  secondary,  as  well  as  the  primary,  and  to  note  that  the 
ratio  of  currents  is  practically  equal  to  the  ratio  of  turns. 

If  there  are  two  secondaries,  it  makes  no  difference  whether  they  be 
put  in  parallel  or  in  series ;  two  primaries  should  be  put  in  parallel  or 
series  to  suit  the  range  of  instruments.  No  coil  should  be  left  idle. 

*  For  example,  eddy-current  loss  in  the  core  due  to  local  flux  set  up  by 
the  current  in  a  loaded  transformer  in  addition  to  the  normal  core  loss. 

t(§i8a).Very  commonly,  however,  this  is  rfot  done,  copper  losses 
(neglecting  load  losses)  being  determined  by  direct  current  measurement 
of  resistance.  This  tends  to  favor  the  transformer.  In  justification  of 
this,  it  may  be  said  that  it  has  not  been  fully  established  that  the  load 
losses  under  actual  load  conditions  are  the  same  as  those  obtained  on 
short  circuit — it  being  held  that  they  may  be  less.  The  two  methods  serve 
as  a  check.  If  the  losses  by  the  wattmeter  are  only  slightly  greater  than 
by  direct  current,  the  result  is  satisfactory  for  the  transformer.  Any  con- 
siderable difference,  however,  shows  the  existence  of  load  losses.  For  an 
accurate  comparison,  great  care  is  necessary  in  regard  to  temperature 
conditions  and  the  calibration  of  instruments. 
12 


162  TRANSFORMERS.  [Exp. 

Although  these  losses  may  be  considerable  in  large  transformers, 
in  small  well  built  transformers  they  are  usually  insignificant. 

§  19.  Impedance  Voltage. — In  an  ideal  transformer  on  short 
circuit,  with  zero  secondary  resistance  and  no  magnetic  leakage, 
the  only  voltage  necessary  to  cause  a  given  current  to  flow  would 
be  RJi,  to  overcome  the  resistance  of  the  primary.  The  effect 
of  secondary  resistance  is  to  apparently  increase  the  resistance 
of.  the  primary  to 

and  the  resistance  drop  is,  accordingly,  RI^. 

On  account  of  the  magnetic  leakage,  the  transformer  appar- 
ently has  a  reactance  X,  called  leakage  reactance;*  this  causes 
(in  terms  of  the  primary)  a  reactance  drop  XIlt  in  addition  to 
the  resistance  drop,  RI^  For  a  given  frequency,  this  reactance 
due  to  leakage  is  a  constant  of  the  transformer,  the  same  as 
resistance.  It  is  the  same  on  open  circuit  or  short  circuit,  and 
is  the  same  at  no  load  or  full  load. 

The  total  impedance,  which  limits  the  flow  of  current  in  the 
short  circuit  test,  is  a  combination  of  the  equivalent  resistance 
and  leakage  reactance,  being 

The  voltmeter  reading  gives  the  total  impedance  voltage, 


necessary  to  overcome  both  resistance  and  leakage  reactance. 

§  20.  Data. — At  rated  frequency,  take,  say,  five  readings  of  the 
impedance  voltage  (Ez)  and  the  copper  loss  (We)  for  various 
currents  from  about  J  to  ij  full-load  current. 

§21.  Readings  at  various  currents  are  chiefly  for  illustration 
and  are  not  essential.  When  facilities  for  varying  the  current 
are  lacking,  one  accurate  reading  (or  better  the  mean  of  five 

*  Discussed  more  fully  in  Exp.  5-C. 


5-B]  TEST  BY  LOSSES.  163 

readings)  at  any  convenient  value  of  current  is  sufficient  for  all 
results.  Slight  changes  in  wave  form  are  immaterial  and  a 
series  resistance  or  any  other  means  may  be  used  for  adjusting 
the  current  (§45). 

§  22.  These  readings  vary  with  temperature,  being  dependent 
upon  resistance,  and  should  be  taken  at  some  definite  tempera- 
ture or  under  some  definite  temperature  conditions  to  be  specified 
in  the  report,  as  hot  after  a  heat  run  of  a  certain  duration,  or 
cold  before  the  transformer  is  heated  up.  In  this  latter  case 
readings  must  be  taken  quickly  to  avoid  rise  in  temperature  due 
to  the  testing  current.  Commercial  tests  should  be  under  speci- 
fied service  conditions,  commonly  after  a  three  hour  heat  run  at 
full  load  or  the  equivalent,  the  room  temperature  being  25°  C. ;  in 
this  the  A.  I.  E.  E.  Standardization  Rules  should  be  consulted 
and  followed. 

§23.  At  a  second  frequency  repeat  the  readings.  (In  a  com- 
mercial test,  readings  would  be  taken  at  rated  frequency  only.) 
It  will  be  found  that  the  copper  loss  and  apparent  resistance 
vary  but  slightly  with  frequency  and  that  the  leakage  reactance 
is  proportional  to  frequency.  Known  for  one  frequency,  it  can 
be  computed  for  any  other. 

§  24.  Results. — The  impedance  voltage,  Ez,  and  the  copper 
losses,  We,  can  be  plotted  directly  from  the  voltmeter  and  watt- 
meter readings,  with  primary  current  as  abscissas,  as  in  Fig.  7 
and  Fig.  8.  It  is  better,  however,  to  proceed  as  follows: 

§  25.  From  the  readings  of  the  ammeter,  voltmeter  and  watt- 
meter in  the  short-circuit  test,  compute,  for  each  observation,  the 
values  of  Z,  R  and  X,  as  given  below,  and  determine  an  average 
value  for  all  the  observations. 

Impedance :         Z  =  EZ-S-I1. 

rent 


five  Resistance :         R  =  We  -f-  ^ 

Reactance :         X  =\/Z2  — 


164 


TRANSFORMERS. 


[Exp. 


These  are  equivalent  or  apparent  values,  in  terms  of  the  primary, 
and  include  the  effect  of  both  primary  and  secondary.  The 
effect  of  load  losses  is  included  in  the  values  of  We  and  R. 


TO- 


£40 


10 


0.25 


0.5  0.75 

AMPERES,  PRIMARY 


1.05 


FIG.  7.     Short-circuit  test,  2  K.W.  transformer,  2,ooo-volt  coil.     Voltage  drop 
due  to  impedance  resistance  and  leakage  reactance. 

§  26.  Compare  this  value  of  R  with  the  value  found  from 
resistance  measurements  of  R^  and  R2  in  §  16. 

§  27.  Where  tests  are  made  at  different  frequencies,  compute 
the  mean  value  of  X--n  for  each  frequency;  the  value  should 
be  about  the  same  for  all  frequencies. 

§  28.  The  values  of  X  affect  regulation  but  not  efficiency ; 
while  the  values  of  R  affect  not  only  regulation  (on  account  of 
RI  drop)  but  also  efficiency  (on  account  of  RI2  loss). 

§  29.  Curves  jor  Voltage  Drop. — Using  the  values  of  Z,  R 
and  X  thus  determined  (§25),  plot  curves  for  Z/1?  RI±  and  XI ^ 
drops  for  different  values  of  primary  current,  as  in  Fig.  7,  these 
curves  being  straight  lines.  Compare  these  curves  with  the  cor- 
responding curves  for  an  alternator,  Fig.  2,  Exp.  3-8. 

For  normal  full-load  current,  mark  the  value  of  each  drop 
in  volts  and  as  per  cent,  of  normal  full-load  voltage.  The  per 


5-B] 


TEST  BY  LOSSES. 


165 


cent,    impedance    drop    is    also    called    impedance    ratio    (§13, 
Exp.  3-B). 

§  30.  Curves  for  Copper  Losses. — Calculate  RI^2  for  -^  J,  ^, 
f,  i  and  ij  load.     Plotted  as  in  Fig.  8,  these  give  the  values  of 


95.38 


121.9 


0.25 


0.5  0.75 

AMPERES,  PRIMARY. 


1.0 


FIG.  8.     Losses  and  efficiency  of  a  2  K.W.  transformer. 

the  copper  loss,  including  load  losses,  for  different  currents;  the 
curve  is  a  parabola. 

It  is  seen  that  the  copper  loss,  in  watts,  for  a  given  load  is  pro- 
portional to  the  copper  drop,  in  volts.  The  copper  loss,  expressed 
as  a  per  cent,  of  rated  volt-amperes,  is  equal  to  the  copper  drop, 
expressed  as  a  per  cent,  of  rated  volts. 

Per  cent,  copper  loss  =  RP/EI.     Per  cent,  copper  drop  =  RI/E. 

PART  IV.    RESULTS.    EFFICIENCY  AND  REGULATION. 

§31.  Efficiency. — Efficiency  is  equal  to  output  divided  by  in- 
put and  is  readily  determined  when  the  losses  are  known.  For 
a  particular  frequency  and  normal  voltage,  take  the  value  of 
core  loss  from  the  curves  already  determined,  Figs.  3,  4  and  5- 


166  TRANSFORMERS.  [Exp. 

Thus,  let  W  0  =  41.  6  watts.  The  total  losses  are  found  by  adding 
this  constant  core  loss  to  the  copper  losses  for  each  load,  as  in 
Fig.  8.  The  efficiency*  should  be  computed  for  j1^,  J,  J,  f  ,  i 
and  ij  load. 

The  computations  for  full  load  and  half  load  are  as  follows  : 
At  -full  load. 

Core  loss      =      41.6 

Copper  loss  =       51.4 

Total  loss     =      93.0 

Output          =2,000.0 

Input  =2,093.0 

-r,  ,  total  loss  93 

Per  cent.  loss=iooX  —  ?  --  :  —  —  100  X  =4.44. 

input  2,093 

^rr-  .  total  loss 

Efficiency  f=  100  —  100  X  —  -•  --  , 

=  100  —  4.44  —  95.56. 
At  half  load. 

Core  loss      =      41.6 
Copper  loss=       12.85 
Total  loss     =       54.45 
Output          =  i  ,000. 
Input  =1,054.45 


Per  cent.  loss  =  100  X  —--—  =  5.1 
1,054.45 

Efficiency  =  i  oo  —  5.16  =94.84. 


*  (§3ia).  The  efficiency  will  be  different  for  different  frequencies  and 
for  different  rating  of  voltage  and  current ;  see  §  48.  See  §  57  for  a  more 
exact  method  of  determining  Wo  for  full-load  voltage. 

Referring  to  Fig.  5,  the  core  loss  at  100  volts  is  41.6  for  the  frequency 
(66.2  cycles)  used  in  the  test.  Corrected  for  60  cycles,  Wo  =  43.3,  it 
being  possible  to  thus  determine  the  efficiency  for  a  frequency  not  used  in 
the  test.  This  is  useful  in  comparing  guarantees. 

t  (§3ib).  This  formula  will  be  found  much  better  for  making  computa- 
tions than  the  equivalent  and  more  usual  form, 

Efficiency  =  Output  -=-  Input. 


5-B]  TEST  BY  LOSSES.  167 

§  32.  Maximum  efficiency*  occurs  at  such  a  load  that  the  cop- 
per loss  is  equal  to  the  core  loss ;  in  Fig.  8,  this  is  at  0.9  full  load. 

Note  the  similarity  between  the  curves  for  a  transformer,  shown 
in  Fig.  8,  and  the  corresponding  curves  for  a  shunt  motor,  Fig.  3, 
Exp.  2-B. 

§  33.  All-day  efficiency  is  computed  on  some  assumption,  as 
5  hrs.  full  load  and  19  hrs.  no  load.  Other  assumptions  can  be 
made  to  suit  specific  service  conditions.  Except  under  special  con- 
ditions, the  term  "  all-day  efficiency  "  has  no  useful  significance. 

§  34.  Regulation. — The  regulation  of  a  constant  potential 
transformer  is  the  per  cent,  increase  in  secondary  voltage  in 
going  from  full  load  to  no  load.  See  Appendix  I.,  Exp.  5~C. 
There  are  various  graphical  methods  for  determining  regulation, 
which  are  necessarily  unsatisfactory  on  account  of  the  small 
values  of  some  of  the  quantities  and  the  consequent  difficulty 
in  making  an  accurate  drawing  to  scale.  There  are  also  various 
analytical  methods,  many  of  which  are  equally  unsatisfactory 
on  account  of  their  involved  character  and  the  unnecessary  labor 
required  in  using  them. 

The  regulation  of  a  transformer  can  be  determined  for  all 
power  factors — current  lagging  or  leading — by  the  same  method 
as  is  used  in  determining  the  regulation  of  an  alternator  by  the 
electromotive  force  method  (§§  16-22,  Exp.  3-6),  either  graph- 
ically or  analytically. 

A  modified  method,  however,  is  easier  to  apply  to  a  trans- 
former on  account  of  the  fact  that  the  resistance  and  reactance 
drops  in  a  transformer  are  comparatively  small. 

§  35.  What  the  writer  believes  to  be  the  simplest  and  most 
practicable  methodf  for  determining  the  regulation  of  a  trans- 

*  See  §  28,  Exp.  2-B. 

t  (§35a).  From  a  paper  "Transformer  Regulation,"  by  F.  Bedell,  Elec. 
World,  Oct.  8,  1898;  the  term  with  «M  is  now  dropped  on  account  of 
difference  of  definition  (see  Appendix  I.,  Exp.  S-C). 


168  TRANSFORMERS.  [Exp. 

former  is  given  below,  any  errors  being  less  than  the  usual 
errors  of  observation. 

Regulation  is  to  be  computed  for  non-inductive  load  and  for 
loads  of  various  power  factors,  with  current  lagging  and  lead- 
ing. (It  is  suggested  that  the  reader  compares  the  results  ob- 
tained by  this  method  and  by  other  methods  with  which  he  may 
be  familiar,  and  that  he  also  compares  the  labor  required  in 
applying  the  different  methods.) 

Let  r  be  the  per  cent,  resistance  drop  and  x  the  per  cent, 
reactance  drop,  as  determined  by  the  short-circuit  test.  Thus, 
in  Fig.  7,  r  =  2.$?  and  ^=1.76  (not  .0257  and  .0176). 

§  36.  Non-inductive  Load. — The  regulation  on  non-inductive 
load  is  computed  as  follows: 

x* 

Per  cent,  regulation  =  r  -I — -. — r-. 

1  2(ioo-(-r) 

For  all  practical  purposes,  as  a  glance  at  the  numerical  ex- 
ample will  show,  this  may  be  written 

A'2 

Per  cent,  regulation  =  r-\ . 

^200 

For  example,  when  r  — 2.57  and  ^-=1.76; 

In-phase  drop—    r   =2.57     per  cent. 

x* 
Effective  quadrature  drop= =  0.015  per  cent. 

Regulation  =  2. 585  per  cent. 

It  is  seen  that  the  regulation  is  practically  determined  by  the 
resistance  drop;  the  effect  of  reactance  drop  on  non-inductive 
load  is  nearly  negligible.  This  is  seen  in  Fig.  9  which  is  dis- 
cussed later.  In  computing  the  regulation,  therefore,  the  accu- 
racy of  the  results  depends  directly  upon  the  accuracy  with 
'which  the  resistance  is  determined.  Regulation  varies  with  tem- 
perature and  to  be  definite  must  be  for  a  specified  temperature. 

§37.  For    Lagging    Current. — When    the    load    has    a    power 


5-B]  TEST  BY  LOSSES.  169 

factor   (cos  6)   less  than  unity  and  the  current  is  lagging,  the 
regulation  is  practically*  as  follows: 

Per  cent,  regulation  =  r  cos  6-\-x  sin  0. 
For  example,  let  cos  0  =  0.866;    sin  0  =  0.5; 

In-phase  resistance  drop  =  r  cos  0  =  2.57X0.866=  2.23  per  cent. 
In-phase  reactance  drop  =  x  sin  0  =  1  .76  X  0.500=0.88  per  cent. 

Regulation  =  3.11  per  cent. 

§38.  For  Leading  Current.  —  When  the  load  has  a  power 
factor  (cos  0)  less  than  unity  and  the  current  is  leading,  the 
regulation  is  practicallyf 

Per  cent,  regulation  =  r  cos  0  —  x  sin  0. 
For  example,  let  cos  0  =  0.866;    sin  0  =  0.5; 

In-phase  resistance  drop  =  r  cos  0^2.57X0.866=2.23  per  cent. 
In-phase  reactance  drop  =-x  sin  0=  1.76  X  0.500  =0.88  per  cent. 

Regulation  =  1.35  per  cent. 

§  39.  Proof.  —  Fig.  9  shows  a  simple  graphical  method  for  obtain- 
ing regulation  at  non-inductive  load.  (Compare  also  Fig.  n,  Exp. 
5~C,  in  which  the  same  lettering  is  used,  and  Fig.  3,  Exp.  3-6.) 

Referring  to  Fig.  9,  lay  off  AL  equal  to  the  secondary  full-load 
voltage  £2=ioo  per  cent.  (A  scale  of  volts  could  be  used,  if 


1-76 

r=2.57 

FIG.  9.  Method  for  determining  regulation  ;  r  =  per  cent,  resistance  drop  ; 
x  =  per  cent,  reactance  drop  ;  regulation  =.  E0  —  E2  =  2.585.  (This  Fig.  is  not 
drawn  to  scale.) 


*(§37a).  For  greater  accuracy,  a  term  for  effective  quadrature  drop 
(<j2-=-20o),  should  be  added,  §42.  In  the  present  .example  this  term  is 
only  .0003,  making  the  regulation  3.1103.  In  any  ordinary  case,  on  lagging 
current,  this  term  can  be  neglected. 

t  (§38a).  For  greater  accuracy,  a  term  g2-=-2OO  should  be  added,  §43; 
in  the  present  example,  this  term  equals  .039. 


1 70  TRANSFORMERS.  [Exp. 

desired,  instead  of  per  cent.)  Lay  off  LJ  =  r  and,  JK  =  x.  Then 
AK  =  E0,  the  secondary  terminal  voltage  at  no  load.  Per  cent,  regu- 
lation =  E0  —  £2  =  2.585. 

As  already  stated,  the  graphical  method  can  not  be  accurately 
applied  on  account  of  the  small  values  of  r  and  x.  From  the 
graphical  method,  an  analytical  method  is  derived  as  follows. 

§  40.  Analytically,  we  have  from  Fig.  9, 

E0  = 
or,  more  simply  (see  §41), 


Transposing,  we  have 

Regulation  =E0  —  i  oo  =  r  -{ -. 

§  41.  Expressed  more  generally,  let 

p  =  in-phase  voltage  drop. 
q  =  quadrature  voltage  drop. 


H0= 


This  practical  identity*  can  be  seen  by  squaring,  or  by  solving  a 
numerical  example.      Transposing,  we  have 


Regulation  =  E,  -  100  =  p  + 

or,  for  practical  purposes,  =  p  -\-  (^2_:_2oo). 

Regulation  =(  in-phase  drop)  -{-(effective  quadrature  drop). 
It  is  the  in-phase  drop  that  chiefly  determines  the  regulation  of  a 
transformer;  effective  quadrature  drop  is  small. 

§  42.  Lagging  Current.  —  For  a  lagging  current,  with  power  factor 
=  cos  0,  the  resistance  drop  r  makes  an  angle  0  with  the  terminal 


*(§4ia).  Theorem. — In  a  right  triangle  in  which  the  height  is  small 
compared  with  the  base,  the  hypotenuse  =  base  -f-  [(height)2-^  2 (base)]. 
This  is  convenient  in  solving  many  alternating  current  problems. 

For  example,  let  base  =100;  when  height  =  5,  hypotenuse  =  100.125 
(true  value  —  100.124922)  ;  when  height  =  10,  hypotenuse  =  100.5  (true 
value  =  100.498756). 


5-B]  TEST  BY  LOSSES.  J7: 

voltage  Er      Fig.  9  would  then  be  drawn  as  Fig.  4,  Exp.  3~B,  in 
which  the  line  BC  makes  an  angle  6  with  the  line  OB. 

Resolve  r  into  an  in-phase  component,  r  cos  0,  and  a  quadrature 
component,  r  sin  6 ;  resolve  x  into  an  in-phase  component,  x  sin  0,  and 
a  quadrature  component,  x  cos  0. 

In-phase  drop  =  p  =  r  cos  0  -f-  #  sin  0. 
Quadrature  drop  =  g  =  .ar  cos  0  —  r  sin  0. 

Regulation  =  ,+  - 


—  p-\-  (g2-f-20o),  approximately. 
For  practical  purposes 

Regulation  =  p  =  r  cos  0  +  x  sin  0. 
§  43.  Leading  Current. — Similarly,  for  a  leading  current, 

p  =  r  cos  6  —  x  sin  6. 
q  =  x  cos  0  -\-r  sin  0. 

Regulation  ==  p  -f- 


2(100  +  p) 
=  p-\-  (g2_:-2oo),  approximately. 

APPENDIX    I. 
MISCELLANEOUS  NOTES. 

§  44.  Selection  of  Instruments. — In  selecting  instruments,  the  neces- 
sary range  can  be  approximately  told  by  assuming  some  reasonable 
value  for  efficiency  or  losses.  Thus,  in  a  2  K.W.  transformer,  let 
us  assume  that  the  efficiency  is  95  per  cent,  and  that  the  iron  losses 
and  copper  losses  are  equal.  Assume  the  power  factor  in  the  open- 
circuit  and  short-circuit  tests  to  be  0.66. 

In  the  open-circuit  test  the  core  loss  will  be  50  watts ;  the  ammeter 
and  current  coil  of  the  wattmeter  must,  accordingly,  carry  a  current 
of  1.5  amperes,  if  the  test  is  made  on  a  5o-volt  coil;  0.75  amp.  for  a 
loo-volt  coil;  0.375  amp.  for  a  2OO-volt  coil,  etc. 

In  the  short-circuit  test,  the  copper  loss  will  be  50  watts;  if  the 


172  TRANSFORMERS.  [Exp. 

test  is  made  on  a  2,ooo-volt  coil,  full-load  current  will  be  I  ampere 
and  the  impedance  voltage  will  be  75  volts;  for  a  i,ooo-volt  coil,  the 
values  become  2  amp.  and  37.5  volts,  etc. 

In  many  cases,  by  a  proper  connection  of  coils  in  series  or  parallel, 
one  set  of  instruments  may  be  selected  which  will  be  suitable  for 
both  open-circuit  and  short-circuit  tests.  It  is  to  be  understood  that 
this  method  of  selection  will  give  only  the  approximate  range.  For 
frequencies  much  below  normal,  it  is  to  be  borne  in  mind  that  the 
current  will  be  much  greater  than  at  normal  frequency  and  in- 
struments with  say  four  or  five  times  the  current-carrying  capacity 
will  be  required. 

§  45.  Adjustment  of  Supply  Voltage. — The  best  way  to  get  various 
voltages  for  the  open-circuit  test  is  by  means  of  a  transformer  or 
an  auto-transformer  with  a  number  of  taps.  An  adjustment  of 
voltage  by  means  of  a  series  resistance  distorts  the  wave  form  and 
hence  introduces  error  in  the  readings  of  core  loss  and  exciting  cur- 
rent. This  error  is  small  if  the  reduction  in  voltage  caused  by  the 
resistance  is  small,  as  from  no  to  104  volts,  and  in  this  case  the  use 
of  series  resistance  is  permissible.  It  should  not  be  used,  however, 
for  large1  reduction  in  voltage  or  in  any  case  when  high  accuracy  is 
wanted.  (In  a  particular  case,*  reducing  the  voltage  from  220  to  no 
volts  by  a  series  resistance  caused  a  decrease  in  core  loss  and  in 
magnetizing  current  of  about  6  per  cent.)  If  a  resistance  is  to  be 
used,  less  error  is  introduced  when  the  resistance  is  bridged  across 
the  line  and  the  transformer  supply  shunted  off  of  part  of  the  resist- 
ance than  is  introduced  with  the  resistance  in  series.  It  is  best,  how- 
ever, to  avoid  resistance  control  entirely. 

Another  way  to  vary  the  voltage  is  to  vary  the  field  excitation  of 
the  supply  alternator.  No  other  load  should  be  on  the  alternator, 
nor  should  the  transformer  under  test  form  an  appreciable  load  on 
the  alternator;  otherwise  change  in  wave-form  may  materially  change 
the  core  loss.  (A  change  of  20  per  cent,  can  be  thus  produced.) 

§46.  The  two-voltage  method  (§9)  obviates  the  necessity  of 
voltage  adjustment.  If  normal  and  half  voltage  are  not  available,  the 
method  may  often  be  used  with  one  voltage  only  by  connecting  coils 

*  This  was  for  an  old  transformer.  With  the  new  transformer  iron 
and  higher  densities,  the  errors  due  to  wave  distortion  become  greater. 


5-B]  TEST  BY  LOSSES.  *73 

first  in  parallel  on  the  appropriate  normal  voltage  and  then  in  series 
on  the  same  voltage  (which  will  then  be  half  normal)  ;  thus,  two 
55-volt  coils  in  parallel  on  a  55-volt  circuit  give  the  reading  for 
normal  voltage,  while  the  two  coils  in  series  on  the  55-volt  circuit 
give  the  reading  for  half  normal  voltage.  For  the  same  degree  of 
magnetization,  the  wattmeter  will  indicate  the  same  core  loss  what- 
ever coil  is  used;  the  ammeter  will  read  twice  as  much  for  the 
parallel  as  for  the  series  arrangement  (the  ampere  turns  being  the 
same)  and  the  ammeter  reading  must,  accordingly,  be  divided  (or 
multiplied)  by  two  to  reduce  all  readings  to  common  terms.  The 
voltmeter  reading  must  be  multiplied  (or  divided)  by  two. 

§  47.  In  the  short-circuit  test,  the  matter  of  wave  form  is  practi- 
cally of  no  consequence ;  the  only  result  affected  is  the  reactance  drop 
and  only  a  very  large  change  in  wave  form  could  materially  affect 
its  value.  For  the  short-circuit  test,  therefore,  any  means  of  adjust- 
ment may  be  used  which  is  found  convenient. 

§  48.  Normal  Voltage  and  Current. — In  determining  normal  full- 
load  values  of  current  and  electromotive  forces,  the  assumption  is 
commonly  made  that  the  efficiency  is  100  per  cent,  and  that  currents 
and  voltages  are  transformed  exactly  in  the  ratio  of  turns.  Thus 
in  a  2,000/100  volt,  2  K.W.  transformer,  the  secondary  current  is 
taken  as  20  amperes  and  the  primary  current  as  I  ampere  (whereas 
strictly  the  latter  should  be  a  trifle  more)  ;  the  secondary  voltage  is 
taken  as  100  volts  and  the  primary  2,000  volts,  neglecting  the  fact 
that  these  no-load  values  of  voltages  do  not  strictly  hold  at  full  load. 
Any  change  of  rating  changes  the  results  of  a  test.  If  the  voltages 
are  rated  as  2,200  and  no,  the  corresponding  primary  and  secondary 
currents  are  0.909  and  18.2  amperes;  the  copper  loss  is  less  and  the 
core  loss  greater.  In  comparing  transformers  and  their  guarantees, 
each  transformer  should  be  tested  at  its  rating.  In  comparing  trans- 
formers for  a  specific  service,  the  tests  of  all  should  be  made  at  a 
common  voltage  according  to  the  conditions  of  the  service.  From  the 
curve  sheets  it  is  very  easy  to  pick  results  for  different  ratings  from 
one  set  of  data.  In  the  laboratory,  to  facilitate  comparison  of  data, 
certain  voltages  should  be  adopted  as  standard,  as  100-200/1,000-2,000, 

1 1 0-220/1, 1 00-2,200,   etc. 

§  49.  Core  Losses  and  Their  Variation. — Eddy  currents  flow  in  local 
short-circuited  secondary  circuits  which  are  practically  non-inductive. 


174  TRANSFORMERS.  [Exp. 

If  e  and  r  represent  the  electromotive  force  and  resistance  of  one  of 
these  circuits,  the  watts  loss  is  ez-±-r.  But  e<xE.  It  accordingly 
follows  that  eddy  current  loss*  varies  as  the  square  of  the  voltage 
and  is  independent  of  frequency  and  wave  form.  As  the  tempera- 
ture of  a  transformer  increases  the  eddy  current  loss  decreases.! 

Referring  to  Fig.  5,  if  all  the  core  loss  were  due  to  eddy  currents, 
we  would  have  b  =  o  and  a  =  2.  The  curve  for  losses  at  different 
frequencies  and  constant  voltage  would  be  a  horizontal  line;  the 
curves  for  losses  at  different  voltages  and  constant  frequency  would 
slope  at  an  angle  with  a  tangent  2. 

§  50.  Hysteresis  loss  in  watts  per  cu.  cm.  is  practically  equal  to 
^M51>6io~7.  Here  ^  is  a  coefficient  of  hysteresis,  equal:}:  to  about  .002 
for  what  was  formerly  good  iron,  but  is  now  little  more  than  half  that 
value  for  the  best  alloy  steel.  The  hysteretic  exponent  1.6,  first 
determined  by  Steinmetz,  is  only  approximate,  being  less  than  this 
for  low  magnetic  densities  and  considerably  more  than  this  for  high 
densities. 

Referring  to  Fig.  5,  if  all  the  core  loss  were  due  to  hysteresis,  we 
would  have  (taking  1.6  as  the  hysteretic  exponent)  0=1.6  and 
b=  i  —  a  =  —  0.6. 

§  51.  In  an  actual  transformer  both  hysteresis  and  eddy  current 
loss  are  present,  so  that  a  has  a  value  between  1.6  (hysteresis)  and  2 
(eddy  currents),  and  b  has  a  value  between  — 0.6  (hysteresis)  and  o 

*(§4Qa).  In  terms  of  B,  eddy  current  loss  in  watts  per  cu.  cm.  is 
y(dnB)2  io~21,  where  d  is  the  thickness  of  lamination  expressed  in  mils 
and  7  is  the  conductance  (the  reciprocal  of  resistance  in  ohms  per  cu. 
cm.)  of  the  material;  for  iron,  7  is  about  io5.  There  is  a  very  slight 
change  of  eddy  currents  with  frequency  and  wave  form  due  to  local 
inductance  and  "  skin  effect "  in  the  local  eddy  current  circuit. 

By  decreasing  the  thickness  of  transformer  plate,  eddy  current  loss  is 
diminished ;  but  hysteresis  loss  is  increased,  since  some  iron  is  wasted  and 
B  is  greater  in  the  remainder.  A  thickness  between  io  and  15  mils  gives 
the  least  total  loss,  according  to  particular  conditions;  see  Elec,  World, 
Dec.  31,  1898. 

t(§4Qb).  Hysteresis  loss,  also,  decreases  with  increase  in  temperature. 
The  total  core  loss  of  a  transformer  when  hot  may  be  6  or  8  per  cent, 
less  than  when  cold. 

\  (§5oa).  By  so  called  "aging  "  due  to  heat,  this  coefficient  increases  in 
the  course  of  time  Although  not  entirely  eliminated,  this  effect  has  been 
reduced  in  the  best  steel  now  used. 


S-B]  TEST  BY  LOSSES.  '75 

(eddy  currents).  Hysteresis  is  the  chief  loss  and  has  most  weight  in 
determining  a  and  b.  Hysteresis  loss*,  and  hence  the  values  of  a 
and  b,  are  affected  by  wave  form.  It  will  be  understood  that  the 
hysteresis  exponent  1.6  is  not  a  constant  but  represents  a  fair  average 
value  for  moderate  ranges  of  flux  densities;  at  high  densities  the 
hysteretic  exponent  may  have  a  value  as  high  as  2  or  more. 

§  52.  One-Foliage  and  One-Frequency  Method. — If  a  and  b  are 
known,  it  is  possible,  having  determined  the  core  loss  at  one  voltage 
for  a  particular  frequency,  to  compute  the  core  loss  for  any  other 
voltage  and  frequency.  It  then  becomes  unnecessary  to  test  a  trans- 
former at  the  exact  rated  voltage  and  frequency — which  is  indeed 
difficult  to  do. 

Taking  as  average  values  a  =1.666  and  b  =  —  .4474,  correction 
factorsf  for  variation  of  core  loss  with  frequency  and  voltage  are 
given  in  the  following  tables. 

CORRECTION  TABLES. 
VARIATION  OF  CORE  Loss  WITH  VOLTAGE. 

Volts  (per  cent,  normal)            90  95  96  97  98  99       100 

Core  loss                                        83.7  91.7  93.3  95  96.7  98.4     100 

Volts   (per  cent,  normal)  101  102  103  104  105  no 

Core  loss  101.6  103.3  105  106.6  108.3  116.6 

VARIATION  OF  CORE  Loss  WITH  FREQUENCY. 

Cycles        55         56  57         5$         59         60    6 1       62       63       64    65 

Loss         103.9    103.12     102.3     101.6     100.8    100    99.3    98.5    97.8    97    96.4 

§  53.  Separation  of  Hysteresis  and  Eddy  Currents. — To  determine 
the  eddy  current  loss  in  watts,  the  core  loss  is.  to  be  measured  at  two 
frequencies  and  at  the  same  flux  density.  Let  W  be  total  core  loss  at 
normal  voltage  E'  and  frequency  n1 '.  At  a  lower  frequency  n"  and  a 

*  (§5ia).  When  the  wave  of  electromotive  force  is  peaked,  the  maxi- 
mum flux  density  and  the  core  loss  are  less. 

t  (§52a).  These  are  taken  from  a  series  of  tests  made  in  1899  by  W.  F. 
Kelley  and  H.  Spoehrer  (see  thesis,  Cornell  University  Library).  The 
transformers  were  small  (1-15  K.  W.)  and  were  designed  for  60  cycles 
and  over.  The  writer  has  no  data  on  most  recent  transformer  iron. 
These  tests  also  showed  that  each  per  cent,  variation  in  voltage  caused 
about  .7  per  cent.  (.6945)  variation  in  exciting  current. 


i76  TRANSFORMERS.  [Exp. 

n" 

lower*  voltage  E"  =  ,  Ef,  let  the  core  loss  be  W".  We  may  com- 
pute the  watts  eddy  current  loss  at  the  higher  frequency  (nr)  and 
normal  voltage  by  the  formula 

W  —  4  W" 
Watts  eddy  currents  =  —          — 


Eddy  current  loss  is  substantially  the  same  for  all  frequencies,  but 
varies  as  the  square  of  the  voltage  and  so  can  be  computed  for  any 
frequency  and  voltage.  Hysteresis  loss  is  found  by  subtracting  eddy 
loss  from  total  loss. 

*  (§  53a).  If  the  wave  form  of  electromotive  force  for  the  two  frequen- 
cies is  different,  E"  =  (n"f  4-  rif'}E',  where  the  form  factor  /  is  the 
ratio  of  the  effective  to  the  average  value.  (For  a  sine  wave,  /=i.i.) 
The  eddy  current  loss  in  watts  at  the  higher  frequency  n'  and  normal 
voltage  is  then 


The  above  equations  can  be  derived  as  follows  :  Eddy  current  loss, 
irrespective  of  frequency  and  wave  form  (§49),  varies  as  E2  and  equals 
aE2,  where  a  is  a  constant.  Similarly,  for  any  wave  form,  hysteresis 
loss  equals  bnB*,  where  b  and  x  are  constant  ;  no  assumption  is  made 
that  x=  1.6.  At  the  two  frequencies  the  total  losses  are 

(1)  W  =  a(E'Y+bn'  (BT; 

(2)  W"  =  a(E"Y  +  bn"(B"Y. 

For  B",  write  B',  this  being  the  condition  of  the  test;  for  E",  write 
E'(E"-^E').  Multiply  (2)  by  n'  -=-  n",  subtract  from  (i)  and  solve 
for  eddy  current  loss  a(E')2.  When  the  wave  form  of  electromotive 
force  is  the  same  at  the  two  frequencies,  (E"  -±-  E'}  =  (M"-T-W')- 

The  separation  of  losses  by  measurements  at  two  frequencies  was  first 
made  by  Steinmetz  ;  the  influence  of  form  factor  was  introduced  by 
Roessler.  There  are  various  methods  for  making  the  calculations,  differ- 
ing somewhat  in  detail.  The  formulae  here  given  are  from  a  paper  by  the 
author  before  the  Cornell  Electrical  Society,  May  4,  1898.  Note  §  35, 
Exp.  5-A,  and  Appendix  L,  Exp.  2-B  ;  also  Bedell's  Transformer,  p.  312 
et  seq.  (Some  of  these  references,  following  Roessler,  use  form  factor 
as  the  reciprocal  of  /,  as  defined  above.)  M.  G.  Lloyd  has  recently  pub- 
lished a  very  complete  investigation  of  the  subject;  see  Bull.  Bureau  oj 
Standards,  February,  1909. 


5-B]  TEST  BY  LOSSES.  i?7 

§  54.  Insulation  and  Temperature  Tests. — These  tests  are  of  com- 
mercial importance  but  need  no  full  discussion  here.  The  Standardi- 
zation Rules  specify  fully  the  conditions  under  which  they  are  to 
be  made;  details  of  the  tests  are  described  in  the  usual  handbooks. 

§  55.  Insulation. — The  insulation  is  tested  between  each  winding 
and  all  other  parts.  The  applied  voltage  is  increased  gradually,  so 
as  to  avoid  any  excessive  momentary  strain.  This  is  usually  done 
by  some  means  of  primary  control  in  a  special  testing  transformer. 
Various  companies  make  testing  transformers  for  obtaining  high 
potential  for  this  test  and  furnish  detailed  instructions  for  their  use. 
The  voltage  is  preferably  measured  by  means  of  a  spark  gap  with 
a  high  protective  resistance  in  series  with  it.  The  test  consists  in 
seeing  that  the  apparatus  withstands  a  specified  over-voltage  for  a 
specified  time  without  breaking  down. 

§  56.  H eat  Runs. — These  are  made  under  full-load  voltage  and 
full-load  current  for  a  specified  time,  temperatures  being  found  by 
thermometers  and  resistance  measurements.  The  heat  run  could  be 
made  by  actually  loading  the  transformer,  but  is  usually  made  by 
some  kind  of  opposition  or  pumping  back  method,  of  which  there  are 
several.  No  load  is  then  required  and  no  power,  except  enough  to 
supply  the  losses. 

A  common  form  of  opposition  run  employs  two  similar  trans- 
formers: the  two  secondaries  (low  potential  side)  are  connected  in 
parallel  to  source  A,  of  normal  frequency  and  normal  voltage,  which 
supplies  the  core  loss;  the  two  primaries  are  connected  in  series, 
opposed  to  each  other,  and  are  then  connected  to  source  B,  which 
supplies  the  normal  full-load  current.  (Source  B  requires  a  voltage 
equal  to  twice  the  impedance  voltage  of  one  transformer  and  can  be 
of  any  frequency,  i.  e.,  it  may  or  may  not  be  the  same  frequency  as 
A.)  All  windings  now  have  full-load  current  and  normal  voltage. 
Instruments  in  A  will  give,  if  desired,  the  core  loss  and  exciting 
current;  instruments  in  B  will  give  copper  loss  and  impedance 
voltage. 

Instead  of  connecting  source  B  in  the  high  potential  side,  a  com- 
mon modification  is  to  connect  the  high  potential  windings  of  the  two 
transformers  directly  in  opposition  and  to  insert  source  B  in  series 
with  the  low  potential  winding  of  one  of  the  transformers.  This  has 

13 


I78  TRANSFORMERS.  [Exp. 

the  advantage  that  all  connections  with  supply  lines  and  instruments 
are  at  low  potential;  see  Electric  Journal,  p.  64,  Vol.  VI.,  and  Fig. 
322,  KarapetofFs  Exp.  Elect.  Engineering. 

A  modified  form  of  opposition  test  can  be  applied  to  a  single 
transformer;  see  Foster's  Handbook. 

§57.  Note  on  Efficiency. — If  the  rated  secondary  voltage  is 
£2=ioo,  the  customary  and  most  simple  procedure  is  to  take  the 
core  loss  for  this  voltage  from  Fig.  5  (thus,  WQ  =  ^i. 6)  and  to 
compute  the  full  load  efficiency  as  in  §  31.  To  be  accurate,  however, 
the  secondary  core  voltage  or  flux  voltage,  Es,  should  be  taken  as 
E2  plus  the  secondary  RI  drop.  Taking  this  drop  as  1.28  (—%r  in 
§35),  we  have  Es  — 101.28  and  the  corresponding  core  loss, 
W0  =  42.5;  this  gives  the  correct  efficiency  of  95.51  instead  of  95.56. 
The  difference  between  these  values  is  so  little  that  the  method  of 
§  31  is  usually  sufficiently  correct. 

Approached  in  another  way,  we  might  consider  EB  =  100  and 
W0  =  4i.6.  Then  £2=100—1.28  =  98.72.  To  get  the  rated  out- 
put of  2  K.  W.,  since  E2  is  decreased,  the  current  must  be  increased 
by  the  factor  I  -f-  98.72.  The  copper  loss  must  then  be  increased  by 
the  factor  (i -=-98.72) 2,  giving  an  efficiency  of  about  95.5. 


5-C]  CIRCLE  DIAGRAM.  179 

EXPERIMENT  5-C.  Circle  Diagram  for  a  Constant  Poten- 
tial Transformer. 

§  i.  Introductory. — It  has  been  seen,  Exp.  4-6,  that  when 
the  resistance  is  varied  in  a  series  circuit  with  constant  reactance 
the  vector  representing  the  current  follows  the  arc  of  a  circle 
as  a  locus.  In  a  similar  manner,  the  primary  current  of  a  con- 
stant potential  transformer  follows  the  arc  of  a  circle  as  a  locus 
when  the  secondary  resistance  is  varied.  The  same  is  true  for 
an  induction  motor  when  its  load  is  varied,  and  use  is  made  of 
this  fact  in  practical  motor  testing.  The  following  experiment 
will,  accordingly,  serve  to  make  clear  certain  principles  of  the 
induction  motor  as  well  as  of  the  transformer;  upon  these  prin- 
ciples is  based  the  method  of  transformer  testing  developed  in 
detail  in  Exp.  5-B. 

In  Part  I.  the  general  principles  governing  the  action  of  a 
transformer  will  be  discussed;  in  Part  II.  these  principles  will 
be  applied  in  constructing  a  circle  diagram.  The  practical  re- 
sults, so  far  as  commercial  testing  is  concerned,  are  all  given 
in  Exp.  5-B.  The  actual  construction  of  a  diagram  to  scale 
gives  one  a  definite  and  concrete  idea  of  what  might  otherwise 
be  vague  and  abstract.  Furthermore,  the  abstract  diagrams 
given  here  (Figs,  i-n)  and  elsewhere  are  so  grossly  exaggerated 
that  they  give  very  wrong  ideas  of  real  values.  Even  Fig.  12, 
which  is  more  nearly  to  scale,  is  much  exaggerated. 

§  2.  Data. — The  same  data  are  required  as  in  Exp.  5-6.  See 
§  25  of  this  experiment. 


PART  I.     GENERAL  DISCUSSION   OF  THE  ACTION  OF  A 
TRANSFORMER. 

§3.  The  action  of  a  transformer  will  be  most  readily  under- 
stood by  considering  its  action  first  without  a  load — i.  e.,  on  open 
circuit — and  then  with  a  load. 


iSo 


TRANSFORMERS. 


[Exp. 


§  4.  Transformer  on  Open  Circuit. — When  a  transformer  is  on 
open  circuit,  the  secondary  winding  has  no  current  flowing  in 
it  and  it  accordingly  has  no  magnetizing  effect  on  the  core. 
A  small  current  flows  in  the  primary  which  magnetizes  the 
core.  Let  us  see  what  determines  the  magnitude  and  phase  of 
this  open-circuit  primary  current. 

§  5.  Assuming  No  Core  Loss. — The  open-circuit  diagram  for 
a  perfect  transformer,  in  which  there  are  no  losses,  is  shown 
in  Fig.  i.  The  primary  electromotive  force  Ep  causes  a  current 
J0  to  flow  and  this  current  sets  up  a  flux  <£.  This  flux,  being 
alternating,  causes  a  counter-electromotive  force  opposed  to  the 
primary  impressed  electromotive  force.  When  the  primary  cir- 
cuit is  closed,  the  current  /0,  and  the  flux  </>  which  it  sets  up, 
assume  such  values  that  the  counter-electromotive  force  is  just 
equal*  to  the  impressed  electromotive  force. 

This  primary  counter-electro- 
motive force  has,  at  any  instant, 
the  value  e'  =  —  S^dQ-t-'dt), 
the  equal  and  opposite  im- 
pressed electromotive  force  be- 
ing eP  =  S1(d<l>^-dt).  It  will 
be  seen  that  the  electromotive 
force  is  zero  when  the  flux 
is  a  maximum  and  that  the  flux 
<£  lags  90°  behind  the  impressed 
electromotive  force  EP,  as  in 
Fig.  i. 

§  6.  In    the    absence    of    core 
loss,  the  current  70  is  in  phase 
with  the  flux  <£,  which  it  produces.     When  permeability  is  constant, 

magnetizing  force  H  is  proportional  to  /0  and  is  in  phase  with 

\ 

*  The  primary  resistance  on  open  circuit  is  very  small  and  can  be 
neglected. 


Flux( 


Phase  of 
~B  and  H 


FIG.  i.     Open-circuit  diagram  for  a 
transformer  with  no  core  loss. 


5-C] 


CIRCLE  DIAGRAM. 


181 


and  proportional  to  the  flux  density  B.  The  B-H  curve  is  a 
straight  line,  instead  of  the  familiar  hysteresis  loop,  and  there  is 
no  hysteresis  loss. 

The  current  /0,  as  shown  in  Fig.  I,  is  in  quadrature  with  the 
electromotive  force  and  is  wattless. 

§  7.  The  flux  <J>  links  with  the  secondary  circuit  and  induces 
in  the  secondary  an  electromotive  force  ES,  lagging  90°  behind 
the    flux.       The    instantaneous 
value  of  the  secondary  electro- 
motive force  is  e$  =  — S2(d<f>-+- 
dt).     It  is  seen  that  Es  is  ex- 
actly  opposite   to    EP   in   phase 
and   is   equal   to  Ep,   multiplied 
by  (S9-*-S& 

§8.  The  flux  <f>  throughout 
this  discussion  refers  to  the  flux 
which  links  with  both  primary 
and  secondary,  and  Ep  and  Es 
are  the  induced  or  flux  voltages,* 
proportional  to  </>.  In  an  ideal 
transformer  there  is  no  other 
flux,  but  in  an  actual  trans- 
former there  is,  in  addition  to 
this  main  flux,  a  relatively  small 
local  or  leakage  flux,  which  links 
with  the  turns  or  part  of  the 
turns  of  one  winding  only  and 
causes  a  reactance  called  leakage 
reactance.  On  account  of  the  drop  due  to  leakage  reactance  and 
the  drop  due  to  the  resistance  of  the  transformer  windings,  as 
discussed  later,  the  terminal  voltages,  E1  and  E2,  are  slightly  dif- 
ferent from  the  flux  voltages  EP  and  E$. 

*  (§  8a) .  Strictly  speaking  Ep  is  not  the  flux  voltage  but  is  equal  and 
opposite  thereto. 


Core-loss  JH 
Component 


'.  r arfA 


O -B 


Magnetising 
Component 


Flux£> 


FIG.  2.  Open-circuit  diagram  for 
a  transformer  with  core  loss,  show- 
ing the  two  components  of  exciting 
current  and  the  angle  a  of  hysteretic 
advance. 


102  TRANSFORMERS.  [Exp. 

§9.  With  Core  Loss. — A  transformer  with  an  iron  core  differs 
from  the  ideal  transformer  just  discussed  because  there  is  a  loss 
in  the  iron  due  to  hysteresis  and  eddy  currents.  The  open- 
circuit  diagram  now  becomes  as  shown  in  Fig.  2.  The  flux 
<f>  is  still  in  quadrature  with  EP  and  Es,  in  accordance  with 
Faraday's  fundamental  law  of  induced  electromotive  force, 
e  —  —  S(d(f>^-dt).  The  exciting  current  70,  however,  can  no 
longer  be  a  wattless  quadrature  current,  for  it  must  have  an 
in-phase  power  component  to  supply  the  core  losses  due  to 
hysteresis  and  eddy  currents.  This  core  loss  component  is 

/H  =  watts  core  loss-^Ep. 

The  exciting  current  70  is,  accordingly,   advanced  in  phase  by 
an  angle  a,  called  the  hysteretic*  angle  of  advance. 

It  is  seen,  therefore,  that  70  consists  of  two  components — 
the  core  loss  component  7n  and  the  true  magnetizing  component 
7n  which  is  wattless  and  in  phase  with  the  flux.  The  total  ex- 
citing currentf  is  the  vector  sum  of  these  two  components : 


§  10.  A  constant  potential  transformer  (one  in  which  EP  is 
constant)  is  a  constant  flux  transformer.  It  therefore  follows 

*  (§9a)-  As  here  defined,  this  angle  includes  the  effect  of  eddy  currents. 

t(§Qb).  The  exciting  current  of  a  transformer  is  distorted,  i.  e.,  has 
a  wave  form  different  from  that  of  the  electromotive  force,  on  account  of 
harmonics  introduced  by  hysteresis.  (See  Appendix  II.,  Exp.  6-A.) 
These  harmonics — currents  of  3,  5,  7,  etc.,  times  the  fundamental  fre- 
quency— are  necessarily  wattless.  They  do  not  appear,  therefore,  in  the 
power  component  In,  but  are  included  in  the  wattless  component  /M. 
Strictly  speaking,  alternating  currents  in  which  harmonics  are  present  can 
not  be  represented  by  vectors  in  one  plane ;  for  practical  purposes,  how- 
ever, the  plane  vector  diagram,  as  here  given,  is  sufficiently  accurate. 
(See  §47,  Exp.  6-A;  also  "The  Effect  of  Iron  in  Distorting  Alternating 
Current  Wave  Form,"  by  Bedell  and  Tuttle,  A.  I.  E.  E.,  Sept.,  1906;  and 
"  Vector  Representation  of  Non-Harmonic  Alternating  Currents,"  by  B. 
Arakawa,  Physical  Review,  1909.)  These  harmonics  have  the  same  value 
at  all  loads;  at  full  load  they  form  such  a  small  part  of  the  total  current 
that  the  distortion  which  they  produce  is  very  small. 


5-C] 


CIRCLE  DIAGRAM. 


/Circle  Locus, 
Primary 


that  70,  /H  and  /M  are  constant,  and  remain  constant  under  all 
loads.  This  would  not  be  quite  true  in  a  transformer  in  which 
the  primary  line  voltage  El  (and 
not  Ep)  is  constant ;  the  difference 
in  the  two  cases  is  very  small. 

§  ii.  Transformer  Under  Load. 
— The  complete  diagram  for  a 
constant  potential  transformer  un- 
der non-inductive  load  is  shown  in 
Fig.  3.  This  will  be  seen  to  be 
exactly  the  same  as  Fig.  2,  the 
open-circuit  diagram,  with  certain 
additions.  As  in  Fig.  2,  we  have 
the  electromotive  forces  EP  and 
E$  opposite  to  each  other  in  phase 
and  in  quadrature  with  the  con- 
stant flux  (f>.  On  open  circuit,  the 
primary  current  /0  flows  as  al- 
ready discussed. 

§  12.  Secondary  Quantities. — 
When  the  secondary  circuit  is 
connected  to  a  load,  a  secondary 
current  I2  flows,  the  value  of 
which  depends  upon  the  load. 
With  non-inductive  load,  this 
current  would  be  in  phase  with 
E$,  if  the  transformer  were 
perfect.  On  account  of  leakage 
reactance,  X2,  the  current  I2 
lags  a  little  behind  E$,  as  shown 
in  Fig.  3.  It  is  to  be  kept  in 
mind  that  Fig.  3  and  other  diagrams  here  given  are  not  at  all  to 
scale,  being  exaggerated  in  order  to  show  more  clearly  the  rela- 
tions between  the  various  quantities. 


/  Circle  Locus  of 
/  Secondary  Current 
/ 

FIG..  3.  Complete  diagram  for 
a  transformer  under  non-induc- 
tive load.  When  a  current  72 
flows  in  the  secondary,  a  load 
current  Ii2)  flows  in  the  pri- 
mary, opposite  in  phase  and  of 
equal  ampere  turns.  72  =  /<2)  X 
ratio  of  turns. 


lS4  TRANSFORMERS.  [Exp. 

The  secondary  terminal  voltage,  E2,  is  a  little  (perhaps  one 
per  cent.)  less  than  E$  on  account  of  reactance  drop  X2I2,  and 
resistance  drop  RJ2,  the  former  in  quadrature  and  the  latter 
in  phase  with  72.  For  a  non-inductive  load,  the  secondary  cur- 
rent, 72,  is  in  phase  with  the  terminal  voltage,  E2.  (For  an 
inductive  load,  72  would  lag  behind  E2  by  an  angle  6,  where  cos  6 
is  the  power  factor  of  the  load.) 

§13.  In  the  secondary,  it  is  seen  that  Es  is  constant  (flux  being 
constant)  and  the  secondary  may,  accordingly,  be  treated  as 
a  simple  constant  potential  circuit.  The  locus  of  the  secondary 
current,  as  the  load  resistance  varies,  is,  accordingly,  the  arc  of 
a  circle,  as  in  any  constant-potential  circuit  with  constant  react- 
ance. (See  Exp.  4-B.) 

§  14.  Primary  Quantities. — It  has  been  seen  that  on  open  cir- 
cuit the  primary  current  assumes  a  certain  value  70,  so  as  to 
produce  a  flux  that  generates  a  counter-electromotive  force  just 
equal  and  opposite  to  the  impressed  electromotive  force.  When 
a  secondary  current  72  flows,  it  disturbs  this  equilibrium  by 
tending  to  demagnetize  the  core.  This  allows  more  current  to 
flow  in  the  primary.  The  primary  current  increases  until  (in 
addition  to  70)  a  current  7(2)  flows  in  the  primary,  the  magnet- 
izing effect  of  which  (ampere  turns)  just  balances  the  magnet- 
izing effect  of  the  current  72  in  the  secondary.  The  magnet- 
izing effect  of  the  secondary  being  thus  neutralized,  the  flux  has 
the  same  constant  value  as  before  (as  though  produced  by  70 
alone),  so  that  the  counter-electromotive  force  produced  by  the 
flux  continues  to  be  just  equal  and  opposite  to  the  impressed  elec- 
tromotive force. 

In  Fig.  3,  the  total  primary  current  Ilt  is  seen  to  be  composed 
of  the  constant  70  (which  is  small)  and  the  load  current  7(2), 
which  is  opposite  to  the  current  72  in  the  secondary  and  equal 
to  72  multiplied  by  ( S2  -=-  S1 ) .  In  a  i :  I  transformer,  the  pri- 
mary load  current  7(2)  is  equal  to  the  secondary  current  72. 


5-C] 


CIRCLE  DIAGRAM. 


185 


§  15.  Fig.  4  shows  that,  in  a  loaded  transformer,  the  resultant 
ampere  turns  are  constant;  hence  the  flux  is  constant,  so  that 
the  counter-electromotive  force — 
irrespective  of  load — equals,  the 
impressed  electromotive  force.  As 
the  load  changes,  the  primary  cur- 
rent assumes  such  a  value  that  the 
resultant  ampere  turns  remains 
constant  and  this  condition  of 
equilibrium  is  maintained. 

§  16.  The  primary  electromotive 
force,  thus  balanced  by  the 
counter-electromotive  force,  is  Ep. 
Referring  to  Fig.  3,  it  will  be 
seen  that  the  terminal  impressed 
electromotive  force,  E15  is  a  little 
greater  (say  one  per  cent,  greater) 
than  Ep,  on  account  of  the  R^^ 
and  XJi  drops,  due  to  primary 
resistance 
ance. 

§  17.  The  locus  of  the  secondary  current  I2  is  the  arc  of  a 
circle  (§13).  Hence  the  locus  of  the  primary  load  current, 
7(2)  in  Fig.  3,  is  the  arc  of  a  circle.  The  total  primary  current, 
/!,  measured  from  O  to  P,  follows  this  same  locus. 

Some  simplified  diagrams  will  now  be  discussed. 

§  1 8.  Representation  of  Transformer  Circuits.  —  From  the 
foregoing  discussion,  it  will  be  seen  that  the  circuits  of  a  trans- 
former may  be  represented  as  in  Fig.  5,  in  which  the  resistance 
and  leakage  reactance  of  the  two  windings  are  considered  as 
external  to  the  transformer.  Furthermore,  the  exciting  cur- 
rent, 70,  is  considered  as  flowing  in  a  shunt  circuit,  also  external 
to  the  transformer.  This  shunt  circuit  consists  of  two  branches : 


FIG.    4.      Diagram    of   ampere 

and    to    leakage    react-  turns;     the    resultant    amPere 

turns  are  constant. 


1 86 


TRANSFORMERS. 


[Exp. 


a  non-inductive  branch  for  the  in-phase  component,  /H,  and  an 
inductive  branch  (without  resistance)  for  the  wattless  quadra- 
ture component  /M.  The  currents  which  would  flow  in  such 
equivalent  shunt  circuits  correspond  exactly  to  the  currents 
/0,  /H  and  /M  which  actually  flow  in  a  transformer. 


Load 


FIG.  5.  Complete  equivalent  of  a  transformer.  The  exciting  current  /„ 
is  considered  as  flowing  in  a  shunt  circuit.  The  resistance  and  leakage  react- 
ance of  primary  and  secondary  are  considered  as  external.  Corresponds  to 
Fig.  3. 

§19.  The  transformer  proper,  in  Fig.  5,  is  considered  as  ideal, 
all  the  losses  being  treated  as  external;  I(2)=I2(S2-^-S1) ; 
and  Ep=:Es(SlL-T-S2).  The  voltage  at  the  primary  terminals, 
E19  is  more  than  Ep  on  account  of  the  drop  in  X±  and  in  R^. 
Likewise,  the  voltage  at  the  secondary  terminals,  E2,  is  less  than 
ES  on  account  of  the  drop  in  X2  and  in  R2. 


*1 


^JWJtr»-A/VV\rt--i 


Load 


•Mi 


FIG.  6.     Equivalent  circuits  as  level  (i  :  i)  transformer.     Corresponds  to  Fig.  7. 

The  total  primary  current  7±  is  seen  to  be  equal  to  the  load 
current  7(2),  plus  (vectorially)  the  small  no-load  current  /0. 

§  20.  Equivalent  Circuits. — The  circuits  of  a  transformer  may 
be  represented  more  simply  by  the  equivalent  circuits  of  Fig.  6, 


5-C] 


CIRCLE  DIAGRAM. 


187 


in  which  all  quantities  are  expressed  in  terms*  of  the  primary. 
This  will  be  most  readily  understood  by  treating  the  transformer 
as  a  "level"  (1:1)  transformer;  we  have  then,  Ep  =  E$;  and 


The  diagram  corresponding  to  Fig.  6  is  shown  in  Fig.  7  and 
is  seen  to  be  the  same  as  Fig.  3  with  all  secondary  quantities 
expressed  in  terms  of  the  pri- 
mary and  drawn  in  the  first 
quadrant. 

§  21.  Simplified  Circuits. — 
The  equivalent  circuits  so  far 
considered  (Figs.  5  and  6) 
and  the  corresponding  diagrams 
(Figs.  3  and  7)  are  prac- 
tically exact  and  may  be  used 
for  the  accurate  solution  of  any 
transformer  problem.  It  will 
be  noted  that  the  resistance 
and  reactance  for  the  two  wind- 
ings are  treated  separately, 
R^XI  in  the  primary  and  R2X2 
in  the  secondary.  By  com- 
bining these  into  a  single  equiv- 
alent R  and  X,  the  trans- 
former circuits  can,  with  little  error,  be  simplified  in  either  of 

• 

two  ways : 

*(§2oa).  To  express  secondary  quantities  in  terms  of  the  primary: 
multiply  current  by  (Sz-^Si)  ;  multiply  voltage  by  (Si-^S2}  ;  multiply  X 
and  R  by  (Si-=-S2)2.  See  §  i6a,  Exp.  5-B.  It  will  be  understood  that 
secondary  quantities  thus  represented  in  the  primary  are  not  the  real 
secondary  quantities  but  the  equivalent  primary  quantities  which  could 
produce  the  same  results ;  thus,  in  a  10 : 1  transformer,  i  ohm  in  the 
primary  is  equivalent  to  o.oi  ohm  in  the  secondary. 

To  express  primary  quantities  in  terms  of  the  secondary,  divide  instead 
of  multiply  by  these  factors. 


FIG.  7.  Exact  diagram  as  level 
transformer,  corresponding  to  Fig.  6. 
The  same  as  Fig.  3  with  secondary 
quantities  expressed  in  terms  of  the 
primary. 


i88 


TRANSFORMERS. 


[Exp. 


i.  All  the  resistance  and  leakage  reactance  are  considered  to 
be  in  the  primary,  as  in  Figs.  8  and  10. 


7? 


i       T     BI     __xi          **       R* 
— t— • — payw\rrTOfflF>^^ 

i  L  Ijmi 


Load 


ami 


FIG.  9.     Simplified  circuits;  R  and  X  all  in  secondary.     Corresponds  to  Fig.  u. 

R  and  X,  this  current  being  either  7t  (as  in  Fig.  8)  or  7(2)   (as 
in  Fig.  9). 

If  70  were  zero,  Figs.  8  and  9  would  not  differ  from  Fig.  6, 
and  all  the  representations  would  be  identical.  In  fact,  70  is 
so  small  that  either  simplification  and  its  resultant  diagram,  Fig. 


FIG.  8.     Simplified  circuits;  R  and  X  all  in  primary.     Corresponds  to  Fig.  10. 

2.  All  the  resistance  and  leakage  reactance  are  considered  to 
be  in  the  secondary,  as  in  Figs.  9  and  n. 

Each  of  these  simplifications  differ  very  little  from  the  more 
exact  representations  already  discussed. 

In  the  actual  transformer,  as  represented  in  Fig.  6,  it  is  seen 
that  the  current  which  flows  through  X2R2  is  7(2),  while  a  dif- 
ferent current  I±  (slightly  larger,  due  to  70)  flows  through  X^R^. 

In  the  simplifications,  the  same  current  is  considered  to  flow 
through  R^X-L  and  R2X2  which  are  now  combined  into  a  single 


5-C] 


CIRCLE  DIAGRAM. 


189 


10  or  n,  may,  for  most  practical  purposes,  be  considered  as 
correct.  This  makes  it  possible  to  use  the  single  equivalent 
values  for  R  and  X  obtained  by  the  short-circuit  test  of  Exp. 
5-B,  and  does  not  require  separate  values 
of  R  and  X  for  the  primary  and  sec- 
ondary circuits. 

§  22.  Again,  the  voltage  which  causes  EI 
I0  to  flow  is  EP,  as  is  seen  in  Fig.  6.  In 
the  simplifications,  this  voltage  is  taken 
as  E2  (Fig.  8)  which  is,  say,  I  per  cent. 
less,  or  as  E^  (Fig.  9)  which  is,  say, 
i  per  cent,  more  than  the  value  of 
Ep  in  the  actual  cases  of  Fig.  6.  This 
would  make  an  insignificant  change 
in  the  value  of  70  which  is  itself 
small.  In  the  latter  case  70  depends 
only  upon  line  voltage  and  is  independ- 
ent of  load. 

§  23.  Diagrams  Compared.— Let  us  FlG>  I0>  simplified  dia- 
compare  the  exact  diagram,  Fig.  7,  with  gram;  R  and  X  all  in  pri- 

,,          .        ,.,-  T^.  mary.   Corresponds  to  Fig.  8. 

the  simplifications,  Figs.  10  and  n. 

In  Fig.  7,  the  primary  and  secondary  RI  drops  are  in  phase 
with  /!  and  7(2),  respectively,  the  XI  drops  being  in  quadrature. 
The  phase  difference  between  7t  and  7(2)  is  small — much  smaller 
in  fact  than  shown  in  the  figure.  The  primary  and  secondary 
drops  may,  accordingly,  be  combined  with  little  error.  This 
may  be  done  by  taking  the  combined  resistance  drop  in  phase 
with  7t  (Fig.  10),  or  in  phase  with  7(2)  (Fig.  n).  The  com- 
bined reactance  drop  is,  in  each  case,  at  right  angles  to  the 
combined  resistance  drop.  In  an  actual  case  little  error  is 
introduced  by  these  simplifications  and  either  may  be  used,  as 
is  most  convenient. 


190 


TRANSFORMERS. 


[Exp. 


PART  II.     THE  CIRCLE  DIAGRAM  AND   ITS  CONSTRUCTION. 

§  24.  The  circle  diagram  for  a  transformer  shows  the  varia- 
tion in  the  primary  current  for  different  values  of  load  resist- 
ance with  constant  impressed  voltage.  In  Fig.  n,  the  primary 


"~  Circle  Locus  ol 
Primary  Current 


Flu** 

FIG.  ii.     Simplified  diagram;  R  and  X  all  in  secondary.     Corresponds  to  Fig.  9. 

current  is  OP,  being  composed  of  the  no-load  current  OA  and  the 
load  current  AP.  As  the  load  resistance  is  decreased  from  in- 
finity to  zero,  the  point  P  will  trace  the  arc  of  a  circle,  and  will 
take  the  position  P"  on  short  circuit.*  If  it  were  possible  to  elimi- 
nate the  resistance  of  the  transformer  windings,  the  point  P 

*  (§243).  If  a  transformer  is  constructed  so  as  to  have  a  large  leakage 
reactance  (or  if  a  reactance  is  included  in  the  circuit  external  to  the 
transformer),  the' short-circuit  current  and  the  diameter,  E^-^X,  are 
reduced.  The  transformer  may  then  be  operated  at  or  near  short  circuit, 
in  which  case  the  current  will  be  nearly  constant.  This  method  is  used 
for  obtaining  constant  current  from  a  constant  potential  line.  (See  §  4a, 
Exp.  5-A.)  Large  reactance  or  magnetic  leakage  in  any  apparatus  tend 
towards  constant  current  operation.  See  §  8,  Exp.  3-A,  §§  27,  273,  Exp. 
3-B,  and  §  14,  Exp.  4-6. 


5-C1  CIRCLE  DIAGRAM.  19 l 

would  complete  the  semi-circle  and  assume  the  position  P'"t  the 
current  in  this  case  (E^-~X)  being  limited  only  by  the  leakage 
reactance,  X.  The  short-circuit  current  of  a  transformer  oper- 
ated at  full  voltage  would  be,  however,  greatly  in  excess  of  the 
carrying  capacity  of  the  transformer  windings,  and,  in  actual 
operation,  the  point  P  does  not  go  far  beyond  the  full-load  point 
P'.  See  also  Fig.  12,  which  is  more  nearly  to  scale. 

§  25.  Data  Necessary. — The  data  necessary  are  the  values  of 
/0,  /H  and  /M,  to  locate  the  point  A,  and  the  leakage  reactance 
X,  to  determine  the  diameter  of  the  semicircle.  f 

These  data  are  obtained   from  the  open-circuit  » 

and  short-circuit  tests  of  Exp.  5-B. 

I  IP' 

All  quantities  are  to  be  in  terms  of  the  pri- 
mary (high-potential)  side;  thus,  in  Fig.  2. 
Exp.  5-B,  the  values  of  70,  /H  and  /M,  meas-  I  pi 
ured  on  the  loo-volt  coil,  are  divided  by  20  to 
obtain  the  corresponding  values  for  the  2,000- 
volt  primary.  This  give  us: 

/0  =  -03025  ;     /H  =  .0208 ;     /M  =  .0220. 

The  reactance  X  for  the  same  transformer,  is 
35.2  ohms;  see  Fig.  7,  Exp.  5-8. 

§  26.  Construction  of  Diagram  from  Experi- 
mental Data. — From  the  data  given  above,  lay 
off  (Fig.  12)  :  °~ 

FIG.  12.     Con- 

„„         ,  ,.,    .         ,  ~    .         r  struction  of  cir- 

OB  =  /H;     BA=IM;      OA=IV  cle  diagram. 

The  diameter  of  the  circle  is  E.L~X  =  2,000-^-35. 2 ^56. 8 
amperes.  The  radius  p  =  E^  -+-  2  X  =  28.4.  These  values  are 
large  compared  with'/0=>.O3  and  full-load  current  7(2)  =  i 
ampere.  It  is,  accordingly,  not  practicable  to  construct  the 
whole  semicircle,  as  in  Fig.  n,  which  is  not  at  all  to  scale. 


192  TRANSFORMERS.  [Exp. 

For  a  working  range  it  can  be  readily  constructed,  as  in  Fig. 
12,  which  is  more  nearly  to  scale,  as  follows: 

Lay  off  A'D  =  I(2)  forTfo,  TV,  J,  J,  f,  i  and  ij  load.  (It  is 
to  be  noted  that,  in  Fig.  12,  the  angle  DAP  is  small  ;  hence  AD 
is  taken  as  practically  equal  to  AP  or  /(2).)  Thus,  for  a  2,000- 
volt,  2  K.W.,  transformer,  AD  is  laid  off,  successively,  equal  to 
.01,  o.i,  .25,  .50,  .75,  i.o  and  1.25  amperes. 

For  each  value  of  AD,  the  point  P  is  located  by  laying  off 


=  p  —  Vf  —  Aff 

which  can  be  derived  from  the  figure  and  is  the  equation  of  a 
circle  referred  to  A  as  an  origin.  The  line  DP  represents  the 
quadrature  component  of  primary  current  due  to  leakage  re- 
actance. This  is  always  small  and  would  be  zero  when  X  =  ot 
for  the  diameter  of  the  semicircle  (see  Fig.  n)  is  then  infinite. 
The  power  component  AD  is,  therefore,  practically  equal  to  AP. 
It  is  to  be  noted  that 


and  OC=OB  +  AD. 
From  these  values,  compute*  for  different  loads 

Primary  current  =  OP  =  VOC+  C?  . 
Power  factor  =OC--  OP. 

The  curves  in  Fig.  4,  Exp.  5~A  were  thus  computed.      Note  . 
§4ia,  Exp.  5-B. 


*  (§26a).  It  will  be  seen,  also,  that 


Watts  input,  ZTi  =  OC  X  £1  ; 
Watts  output,  W-i  =  Wi  —  losses  ; 


This  gives  a  possible  method  for  determining  the  total  voltage  drop. 


5-C]  CIRCLE  DIAGRAM.  J93 

APPENDIX    I. 
NOTE  ON  REGULATION. 

§  27.  Definition  of  Terms. — Regulation  is  defined  by  the  Institute 
as  follows: 

In  constant-potential  transformers,  the  regulation  is  the  ratio  of  the  rise 
of  secondary  terminal  voltage  from  rated  non-inductive  load  to  no  load 
(at  constant  primary  impressed  terminal  voltage)  to  the  secondary  ter- 
minal voltage  at  rated  load.  (Compare  §34,  Exp.  5-B.) 

If  the  secondary  terminal  voltage  is  EQ  at  no  load  and  E2  at  full 
load,  the  regulation  is, 

Regulation^  (E0  —  E2)  -±-E2 

The  drop  on  which  regulation  depends  is  E0  —  £,,  which  we  may 
term  the  regulation  drop.  This  drop,  expressed  as  per  cent,  of  E2, 

i   gives  the  regulation. 
§  28.  The   total  voltage  drop,  in  terms  of  a   I :  I   transformer,  is 
E^  —  E2  and  is  a  little  more  than  the  regulation  drop,  because  El  is 
a  little  more  than  E0  on  account  of  the  drop  due  to  exciting  current 
in  the  primary  winding. 

The  per  cent,  voltage  drop  is  (E1  — £2)  -=-£.,,  taking  E2  as  100  per 
cent.;  or,  (£,  —  £,)  -t-Elt  taking  El  as  100  per  cent. 

§  29.  Numerically,  the  difference  between'  "  regulation  "  and  "  per 
cent,  voltage  drop  "  is  small.  In  earlier  usage,*  the  term  regulation 
was  commonly  employed  to  designate  "  per  cent,  voltage  drop."  This 
confusion  is  one  cause  for  the  differences  between  various  methods 
which  have  been  used  (and  still  are  used)  for  determining  regulation. 
A  difference  arises,  also,  according  to  whether  £t  or  E2  is  taken  as 
100  per  cent. 


*(§2ga).  See  the  following  articles,  in  the  Elec.  World,  on  the  pre- 
determination of  transformer  regulation :  Bedell,  Chandler  and  Sherwood, 
August  14,  1897;  A.  R.  Everest,  June  4,  1898;  F.  Bedell,  October  8,  1898. 
See  also  Foster's  Electrical  Eng.  Pocket  Book,  p.  492,  fifth  edition,  1908. 

14 


194  TRANSFORMERS.  [Exp. 

§  30.  An  illustration  will  make  this  clear.     Let 
£,=  100;  £0  =  99.9;  £2  =  97- 

Regulation  drop  —  £0  —  £.,  =  2.9  volts. 
Per  cent,  regulation  =  2.9  -=-  97  =  2.99  per  cent. 
Total  voltage  drop  —  £t  —  £2  =  3  volts. 
Per  cent,  voltage  drop  =  3-^-100  =  3  per  cent.;  or 
=  3  -+-  97  =  3-1  Per  cent- 

§  31.  Regulation  drop  depends  upon  the  difference  between  £0  and 
£2.  This  drop  is  due  to  load  current,  and  does  not  include  any  drop 
due  to  exciting  current,  which  affects  E0  and  E2  alike  and,  practically, 
does  not  affect  their  difference. 

The  total  voltage  drop  depends  upon  the  difference  between  El  and 
£2.  This  drop  is  chiefly  due  to  load  current,  but  includes,  in  addition, 
a  small  drop  due  to  exciting  current  which  affects  £2  but  not  £x  and 
so  directly  affects  their  difference. 

§  32.  Computations.  —  To  compute  regulation  drop,  we  have  the 
problem:  Given  £,,  to  compute  E0. 

To  compute  total  voltage  drop,  we  have  the  problem:  Given  £2,  to 
compute  £a. 

§  33.  Regulation.  —  For  determing  regulation,  we  compute 

E0  =  V  T£7+>T~+?  ;  where 

in-phase  drop  =p  =  RI(2), 
quadrature  drop  =.q  =  XI(2). 

It  is  seen  that  exciting  current  does  not  enter.  The  various  drops 
may  be  expressed  either  in  volts  or  in  per  cent.  The  working  details 
of  the  method  are  discussed  in  §§  34-43,  Exp.  5~B. 

§  34.  Total  Voltage  Drop.  —  For  determining  total  voltage  drop,  we 
compute 


The  in-phase  drop  p,  consists  principally  of  /?/(2),  but  includes  the 
small  additional  terms  RJu  and  XJu,  which  are  drops  caused  by  the 
two  components  of  the  exciting  current  flowing  through  Xv  Rr 
Without  much  error,  Xv  R^  may  be  taken  as  half  of  X,  R.  Hence 


5-C]  CIRCLE  DIAGRAM.  195 


In  a  like  manner 

q  =  XI(2)  4-  -^/H  — 


The  last  two  terms  are  small  and  nearly  cancel  each  other. 

§  35.  Other  methods  of  analysis  may  be  employed  for  determining 
the  total  voltage  drop,  and  the  form  in  which  the  results  are  ex- 
pressed will  vary  according  4:o  the  manner  in  which  the  various  terms 
are  combined  and  the  approximations  which  are  introduced.  In  any 
case  some  small  and  troublesome  terms  are  introduced,  which  affect 
the  result  very  little  and  which  do  not  enter  in  the  determination  of 
"  regulation,"  as  denned  by  the  Institute.  The  results  are  affected 
less  by  these  small  terms  than  by  variations  in  the  value  of  R,  depend- 
ing upon  whether,  in  its  determination,  load  losses  were  included  or 
not,  and  whether  steady  temperature  conditions  were  maintained 
during  the  test. 

§  36.  It  might  well  be  held  that  regulation  should  be  so  denned  that 
magnetising  current  should  enter  into  its  determination,  particularly 
since  magnetising  current  has  been  much  increased  by  the  use  of 
improved  iron  worked  at  'higher  densities;  on  the  other  hand,  it  is 
much  simpler  to  define  regulation  independently  of  magnetizing  cur- 
rent and  to  specify  the  value  of  the  magnetizing  or  exciting  current 
as  a  separate  item. 


CHAPTER   VI.. 
POLYPHASE   CURRENTS. 

EXPERIMENT  6-A.  A  General  Study  of  Polyphase  Cur- 
rents.* 

PART   I. 

§  I.  Introductory. — In  a  polyphase  system,  several  single- 
phase  currents  differing  in  phase  from  each  other  are  combined 
into  one  system.  The  circuits  for  each  phase  may  be  independ- 
ent, without  electrical  connection,  or  interconnected.  The  phase 
difference  between  the  currents  of  the  several  phases  is  usually 
90°  or  120°,  the  corresponding  systems  being  called  two-phase 
or  three-phase. 


(a) 

FIG.  i.  Two-phase  connections  for  generator  or  receiver  circuits,  a,  4-wire 
system  with  independent  phases.  b,  3-wire  system.  c,  Quarter-phase,  star- 
connected  ;  or,  4-wire  system  with  interconnected  neutral,  d,  Quarter-phase, 
mesh-connected  ;  or,  ring-connected. 

To  form  a  polyphase  system  we  must  have  several  sources  of 
single-phase  electromotive  force  which  differ  in  phase  by  proper 
amounts.  For  a  symmetrical  polyphase  system  these  electro- 
motive forces  must  be  equal  and  differ  from  each  other  by  equal 
phase  angles,  as  in  the  3-phase  and  quarter-phase  systems  soon 

*(§ia).  In  making  polyphase  measurements,  some  form  of  voltmeter 
and  ammeter  switches  will  be  found  convenient,  so  that  all  readings  can  be 
made  with  one  voltmeter  and  one  ammeter.  The  same  switches  will  serve 
to  transfer  one  wattmeter  from  one  circuit  to  another. 

196 


6-A] 


GENERAL   STUDY. 


!97 


to  be  discussed.  The  sources  of  these  electromotive  forces  are 
in  principle  several  rigidly  connected  single-phase  generators, 
but  in  practice  they  are  generator  coils  on  a  single  armature. 
The  secondary  coil  of  a  transformer  may  be  considered  as  a 
generator  coil.  The  currents  from  these  sources  may  be  utilized 
separately  as  single-phase  currents  (as  in  lighting),  or  jointly  as 
polyphase  currents  (as  in  an  induction  motor). 


(<*) 

FIG.  2.  Three-phase  connections  for  generator  or  receiver  circuits,  a,  Inde- 
pendent circuits  ;  see  §  3a.  b,  Star-  or  F-connected.  c,  Mesh-  or  delta-  (A) 
connected.  d,  T-connected.  e,  F-connected ;  or,  open  delta. 

§  2.  The  load  on  a  polyphase  system  is  balanced  when  each 
phase  has  an  equal  load  with  equal  power  factor.  In  a  balanced 
polyphase  system  the  flow  of  energy  is  uniform,  which  is  a  bet- 
ter and  more  general  definition  of  such  a  system;  (see  Steinmetz, 
Alternating  Current  Phenomena).  In  a  single-phase  system  or 
unbalanced  polyphase  system,  the  flow  of  energy  is  pulsating, — 
discussed  further  in  §  I,  Exp.  7-A.  The  torque  is,  accordingly, 
pulsating  in  all  single-phase  machinery;  whereas  it  is  uniform  in 
polyphase  motors  and  in  polyphase  generators  on  balanced  load. 
Furthermore,  a  polyphase  induction  motor  on  account  of  its 
rotating  field  can  be  given  a  good  starting  torque,  whereas  a 
single-phase  induction  motor  has  none  in  itself  and  has  only  a 
small  starting  torque  when  auxiliary  starting  devices  are  used. 
Polyphase  machinery  has  a  greater  output  than  single  phase  for 
a  given  size,  or  has  a  smaller  size  for  a  given  output.  These 
features,  together  with  the  copper  economy  of  3-phase  as  com- 
pared with  single-phase  transmission,  all  favor  the  use  of  poly- 
phase systems ;  see  Appendix  III. 


I9S  POLYPHASE   CURRENTS.  [Exp. 

§  3.  Methods  of  Connecting  Phases. — Generating  or  receiving 
coils  or  circuits  may  be  combined  in  various  ways,  the  common 
ones  being  shown*  diagrammatically  in  Figs.  I  and  2.  In  Figs. 
I  and  2,  the  relative  positions  of  the  various  coils  represent  the 
relative  phase  positions  of  their  several  electromotive  forces.f 
The  black  dotsj  may  be  taken  as  line  wires  in  cross-section.  On 
paper  the  distance  between  any  two  dots  is  the  difference  of 
potential  between  them ;  phase,  as  well  as  magnitude,  is  shown  in 
this  way. 

To  the  same  polyphase  system,  a  number  of  differently  con- 
nected polyphase  generators  and  receivers  may  be  connected  at 
the  same  time ;  thus,  on  a  3-phase  system,  some  apparatus  may  be 
delta-  and  some  star-connected.  From  a  4-wire  2-phase  system, 
induction  motors  may  be  run  simultaneously  when  connected  as 
(a),  (c)  or  (d),  Fig.  i.  Connection  (b)  can  be  combined  on 
the  same  system  with  (a),  but  not  with  (c)  or  (d).  This  is  an 
objection  to  3~wire  2-phase  distribution,  inasmuch  as  synchron- 
ous motors  and  converters  as  well  as  generators  are  frequently 
wound  quarter-phase  and  so  cannot  be  run  from  a  3-wire  system. 
A  further  objection,  that  the  line  drop  in  the  common  wire  makes 
the  voltages  unsymmetrical,  is  discussed  later,  §  14. 

§4.  Object. — In  performing  this  experiment,  the  object  is  to 
gain  a  knowledge  of  the  connections  of  polyphase  circuits  and 
polyphase  apparatus,  and  to  understand  their  electrical  relations 
and  various  diagrammatic  methods  for  representing  them. 
Make  a  study  of  whatever  polyphase  supply  circuits  are  available 
and  by  means  of  transformers  obtain,  so  far  as  possible,  all  the 
systems  indicated  in  Figs.  I  and  2. 

*  (§3a)-  The  arrangement  of  Fig.  2  (a)  is  never  used  for  independent 
3-phase  circuits ;  it  is  used  only  for  connecting  transformer  secondaries  to 
so-called  6-phase  synchronous  converters,  §  27. 

f(§3b).  Although  the  diagram  of  connections  can  not  in  general  be 
taken  as  the  vector  diagram  of  electromotive  forces,  this  can  be  done  in 
the  simpler  cases  and  makes  the  introduction  to  the  subject  more  clear. 

$  (§3c).  This  representation  by  dot's  is  called  by  Steinmetz  the  topo- 
graphic method. 


6-A] 


GENERAL   STUDY. 


199 


Note  also  the  connections  on  various  pieces  of  polyphase 
apparatus  (as  generators,  motors,  etc.)  which  may  be  available, 
and  note  for  what  kind  of  polyphase  system  the  apparatus  is 
intended. 

PART    II. 

§  5.  Two-phase  Measurement. — Take  two  transformers*  with 
the  same  ratio  of  transformation  (say  1:1).  Connect  the 
primary  of  one  transformer  to  phase  A  of  a  2-phase  circuit,f 
and  the  primary  of  the  other  transformer  to  phase  B.  Measure 
the  secondary  voltages  when  the  secondary  circuits  are  inde- 
pendent, thus  forming  a  4~wire  system  with  independent  phases, 
Fig.  i  (a). 

§  6.  Addition  of  Electromotive  Forces. — Connect  the  two  sec- 
ondaries as  a  3-wire  system,  Fig.  I  (&),  and  measure  the  voltage 
of  each  phase  (£A  and  EB)  and  the  voltage  E  between  outside 
wires.  Lay  off  these  voltages  as  a  triangle  and  note  how  nearly 
EA  and  EE  are  at  right  angles,  so  making  a  true  2-phase  system. 
This  triangle  may  be  drawn  as  in  Fig.  3,  4  or  5. 


FIG.  3.     Topographic 
method. 


FIG.  4.    Addition 
method. 


FIG.  5.     Subtraction 
method. 


§  7.  If  we  use  the  topographic  method  of  Steinmetz  and  omit 
arrows,  we  can  represent  the  electromotive  forces  of  the  2-phase 
3-wire  system  by  Fig.  3  (see  Appendix  I.).  This  electromotive 

*  It  is  preferable  that  each  secondary  consists  of  two  equal  coils :  thus, 
we  might  have  primary  no  volts;  secondaries  55  volts  each,  giving  in 
series  no  volts  with  a  middle  or  neutral  point.  Note  the  various  possible 
voltage  transformations  for  each  transformer. 

t  It  matters  not  whether  the  supply  circuit  is  3-wire  or  4-wire,  or  how 
connected.  If  several  kinds  of  supply  circuits  are  available,  use  each  one 
in  turn.  Compare  Fig.  6. 


200  POLYPHASE   CURRENTS.  [Exp. 

force  diagram  is  seen  to  be  similar  to  the  circuit  diagram  (b)  of 
Fig.  i. 

§  8.  If  we  take  one  outside  line,  say  B,  as  our  starting  point 
(imagining  if  we  wish  that  it  is  grounded,  but  this  is  unneces- 
sary), we  have  the  electromotive  forces  EE  and  £A  represented 
by  the  vectors,  BO  and  OA,  in  the  direction  shown  by  arrows  in 
Fig.  4.  The  sum  of  these  two  vectors  is  BA. 

§  9.  If,  as  is  common,  we  take  the  neutral  O  as  the  starting 
point  (say  ground),  the  differences  of  potential  between  the  side 
wires  and  ground  are  OA  and  OB,  the  direction  of  the  vectors 
being  from  the  starting  point  as  in  Fig.  5.  The  difference  in 
potential  between  A  and  B  is  now  the  difference  between  OB 
and  OA,  Fig.  5;  which  is  the  same  as  the  sum  of  BO  (equal 
-OB)  and  OA,  Fig.  4. 

§  10.  In  general,  if  we  take  electromotive  forces  in  sequence — 
as  BO,  OA — they  must  be  added  (Fig.  4)  ;  if,  however,  we  con- 
sider each  electromotive  force  in  a  direction  away  from  a  com- 
mon joining  point — as  OB,  OA — they  must  be  subtracted 
(Fig.  5).  For  the  simple  case  at  hand,  involving  only  two  elec- 
tromotive forces  connected  to  a  common  point,  the  difference 
method  may  be  readily  applied.  For  more  complicated  networks 
the  addition  method  is  used,  as  it  is  capable  of  more  general 
application ;  it  is  based  on  the  statement  of  Kirchhoff 's  Law 
(§  32,  Appendix  I.)  that  the  differences  of  potential  around  any 
mesh  add  up  to  zero.  For  this  addition  method,  all  arrows  are 
taken  consecutively,  from  feather  to  tip. 

§  ii.  If  each  transformer  has  a  secondary  winding,  consisting 
of  two  equal  coils,  connect  the  secondary  coils  of  the  two  trans- 
formers so  as  to  form  a  star-connected  and  a  mesh-connected 
quarter-phase  system,  as  in  (c)  and  (d)  of  Fig.  I.  Measure 
all  voltages  and  draw  diagrams  of  voltages  for  the  star  and  for 
the  mesh  connection. 

In  the  mesh  connection,  the  two  secondaries  of  one  trans- 
former are  connected  as  the  opposite  sides  of  a  square,  -due 


6-A]  GENERAL   STUDY.  201 

attention  being  given  to  polarity;  the  two  secondaries  of  the 
other  transformer  form  the  remaining  two  sides.  Before  clos- 
ing the  square,  connect  a  voltmeter  between  the  two  points  about 
to  be  connected  and  proceed  to  connect  them  only  in  case  the 
voltmeter  reads  zero.  This  precaution  should  be  taken  in  mak- 
ing any  mesh  connection. 

§  12.  A  convenient  laboratory  supply  board  is  obtained  from 
2-phase  secondaries,  the  secondary  circuit  on  each  phase  consist- 
ing of  four  equal  coils  in  series  so  as  to  form  a  5-wire  system 
on  each  phase.  With  the  neutrals  of  the  two  phases  intercon- 
nected, this  gives  supply  voltages,  as  Fig.  6.  If  the  total  volt- 
age of  each  phase  is  220  volts,  this  gives 
2-phase  voltages  as  follows:  4-wire  no 
and  220  volts;  3~wire  55,  no,  77.8  and 
155.6  volts;  also  additional  single-phase  *i  °i 
voltages,  123  and  165  volts.  The  voltage 
between  any  two  points  can  be  scaled  off 
from  the  drawing  in  Fig.  6,  as  shown  in 
the  discussion  of  Figs.  3,  4  and  5.  When  FlG-  6.  Two-phase 

,  ,  j  ,    ,  laboratory   supply  volt- 

the  transformer  secondaries  cannot  be  so 

ages. 

subdivided,  the  result  can  be  obtained  by 

connecting  across  each  phase  of  a  4-wire  system  an  autotrans- 
former  made  of  four  equal  coils.  Verify  these  voltages  by  calcu- 
lation or  by  measurement. 

The  preceding  study  has  brought  out  the  fact  that  in  poly- 
phase circuits,  the  single-phase  voltages  of  interconnected  gen- 
erator or  receiver  coils  are  combined  geometrically  to  give  re- 
sultant voltages.  Although  this  was  shown  particularly  for 
2-phase  circuits,  it  will  be  understood  to  be  general  and  to  apply 
as  well  to  a  3-phase  circuit  or  to  any  circuit  whatsoever. 

§  13.  Addition  of  Currents. — Currents,  also,  when  of  differ- 
ent phases,  are  added*  vectorially  to  obtain  the  resultant  current. 
To  show  this  proceed  as  follows: 


*  Branch  currents,  flowing  to  or  from  a  common  point,  always  combine 
by  addition — not  by  subtraction — to  give  the  total  current.     See  Appendix  I. 


202 


POLYPHASE   CURRENTS. 


[Exp. 


From  a  3-wire  2-phase  supply,  connect  two  resistances  as  load, 
one  on  each  phase.  Measure  the  currents,  I  A  and  IE,  in  each 
resistance  and  the  total  current  /  in  the  common  conductor.  If 
the  two  currents  I  A  and  IE  differ  in  phase  by  90°,  we  will  have 
/=  V^A2  +  /B2.  This  will  be  true  for  an  inductive  as  well  as 
A  for  a  non-inductive  load,  provided  the 

load    on    each    phase    has    the    same 
power  factor, — i.  e.,  OA  =  OE. 

If  EA  and  EE  are  not  at  right 
angles,  or  OA  and  OE  are  not  equal,  the 
currents  I  A  and  IE  will  no  longer  be  at 
right  angles;  the  branch  currents  will 
still,  however,  add  as  vectors  to  give 
the  total  current,  as  in  Fig.  7. 

§  14.  Line  drop. — To  illustrate  line 
drop,  with  the  same  circuits  and  re- 
sistances just  used,  insert  a  small  additional  non-inductive  resist- 
ance in  the  supply  wires  to  represent  resistance  in  a  long  supply 
line. 

Construct  a  triangle  OAB  for  the  supply  voltage  and  O'A'B' 
for  the  delivered  voltage  for  the  following  three  cases: 

A 
A' 


FIG.  7.     Addition  of  currents. 


O    IB 


B'     B 


FIG.   8.      Resistance   in        FIG.   9.      Resistance   in      FIG.    10.      Resistance   in 


lines  A  and  B. 


common   conductor. 


all   three  lines. 


With  resistances  in  lines  A  and  B  only,  Fig.  8; 
With  a  resistance  in  the  common  conductor  O  only,  Fig.  9; 
With  resistances  in  all  three  lines,  Fig.  10;  in  this  third  case 
measurements  of  voltages  O'A  and  O'B  are  also  to  be  taken. 
For  the  first  case  (Fig.  8),  the  supply  voltages,  OA  and  OB, 


6-A]  GENERAL   STUDY.  203 

shrink  to  the  delivered  voltages,  OA'  and  OB'  \  the  drop  due  to 
resistance  in  lines  A  and  B  is  in  phase  with  the  currents  /A  and 
IE-  There  is  the  same  phase  difference  (90°)  between  the  deliv- 
ered voltages  as  between  the  supply  voltages. 

For  the  second  case  (Fig.  9),  the  line  drop  in  the  common 
conductor  is  in  phase  with  I,  and  it  is  seen  that,  on  account  of 
this  drop,  the  phase  angle  between  the  delivered  voltages  is 
greater  than  between  the  supply  voltages.  This,  also,  is  true  in 
Fig.  10.  This  lack  of  symmetry  in  delivered  voltages  is  one  dis- 
advantage of  the  3-wire  system;  see  §  3. 

§  15.  These  diagrams  illustrate  the  topographic  or  mesh 
method  for  representing  electromotive  forces.  The  direction 
assigned  to  any  line  depends  upon  the  sense  in  which  it  is  taken. 
Resistance  drop  consumed  by  resistance  is  in  phase  with  cur- 
rent; resistance  drop  produced  by  a  resistance  is  opposite  to  the 
current,  as  discussed  in  Exp.  4~A.  It  is  taken  in  this  latter  sense 
in  applying  the  mesh  principle, — Law  (i)  of  Appendix  I. — that 
the  electromotive  forces  around  any  mesh  have  a  vector  sum  of 
zero  and  can  be  represented  as  a  closed  polygon.  Thus,  in 
Fig.  10,  proceeding  around  the  mesh  OAA'O',  we  have  the  fol- 
lowing electromotive  forces :  OA  produced  by  the  generator ; 
AA'  produced  by  resistance  in  line  A  and  opposite  to  /A;  A'O' 
the  counter  electromotive  force  produced  by  the  load  (the  elec- 
tromotive force  delivered  to  the  load  being  O'A')  ;  O'O  pro- 
duced by  resistance  in  the  common  line  O  and  opposite  to  7. 
The  line  drop  for  a  single-phase  circuit  can  be  similarly  repre- 
sented. 

With  inductance  in  the  lines,  besides  the  resistance  drop  just 
discussed,  there  is  a  reactance  drop  at  right  angles  to  the  cur- 
rent; this  reactance  drop  is  90°  ahead  of  the  current  when  con- 
sidered as  consumed  by  reactance,  and  90°  behind  the  current 
when  considered  as  produced  by  reactance.  (See  §  i8a  and  Fig. 
2,  Exp.  4-A,  and  Figs.  3,  4  and  5,  Exp.  3-B.) 

§  1 6.  The  line  drop  diagram,  Fig.  10,  is  true  for  any  3-wire 


204 


POLYPHASE  CURRENTS. 


[Exp. 


system,  and  may  be  applied  to  a  3-phase  system  by  making  the 
triangles  more  or  less  equilateral.  A  4-wire  system  or  any  other 
system  can  be  treated  in  a  similar  manner. 

Furthermore,  the  method  just  discussed  for  treating  the 
effect  of  resistance  drop  and  reactance  drop  in  line  conductors 
is  not  limited  to  non-inductive  loads,  but  is  applicable  as  well  to 
other  loads,  either  with  leading  or  lagging  currents.  (See  §  56. 
Exp.  3-B.) 

§  17.  Conclusion. — In  the  main  it  has  been  seen  that  2-phase 
circuits  are  essentially  the  same  as  two  single-phase  circuits  and 
can  be  so  treated.  Three-phase  circuits  are  likewise  essentially 
three  single-phase  circuits  and  the  conception  of  polyphase  cir- 
cuits is  thus  made  simple.  In  any  polyphase  circuit  the  funda- 
mental principles  for  the  vector  addition  of  currents  and  electro- 
motive forces  apply  as  in  single-phase  circuits.  For  3-phase 
circuits,  however,  there  are  modified  forms  of  treatment  that  are 
found  practically  convenient;  these  will  now  be  considered. 

PART   III. 

§  1 8.  Three-phase  Measurement. — The  most  important  3-phase 
connections  (Fig.  2)  are  the  star  and  delta  connections,  the  elec- 
trical relations  of  which  will 
first  be  studied.  Other  3-phase 
connections  will  then  be  stud- 
ied with  reference  to  various 
arrangements  of  transformers 
on  3-phase  circuits. 

§  19.    Star-connection. — On 
a  3-phase  line,  connect  three 
approximately     equal     resist- 
ances* in  star-connection ;  see 
Fig.    ii.       Measure   the    line 
voltages  XY,  YZ  and  ZX;  these  are  also  called  delta  voltages 
*  Some  measurements  should  also  be  made  with  unequal  resistances. 


Line  Vohtage 


FIG.   ii.     Star-  or  y-connection  of 
load  resistances. 


6-A] 


GENERAL   STUDY. 


205 


and  for  clearness  may  be  designated  by  the  subscript  D  —  thus, 
ED.  When  nothing  further  is  specified  than  the  voltage  E  of 
a  3-phase  line  or  machine,  it  is  this  delta  or  line  voltage  that 
is  meant. 

Measure*  the  star  voltage  Es  (called  also  voltage  per  phase 
or  phase  voltage,  §  30)  from  each  line  to  the  junction  O,  Fig.  n. 
Also  measure  the  star  current  /s  for  each  phase.  The  line  cur- 
rent is  always  the  star  current,  as  is  evident  for  this  case. 

Compare  the  measured  values  of  ED  and  ES  with  the  expres- 
sion (which  should  be  proved) 


§  20.  Compute  the  power  for  each  resistance.  This  is  obvi- 
ously, as  in  a  single-phase  circuit,  equal  to  the  product  of  volts  X 
amperes  (for  a  non-inductive  load), 
i.  e.,  the  product  of  star  voltage  and 
star  current  (£s/s)  for  each  phase.  For 
an  inductive  load  in  which  the  current 
lags  by  an  angle  6,  as  in  Fig.  12,  the 
power  for  each  star  circuit  is  Es  Is  cos  0. 
When  ES,  Is  and  6  are  the  same  for  each 
phase,  we  can  multiply  the  power  for 
each  phase  by  3  to  obtain  the  total 
power;  thus, 


Total  power  — 


But 


hence 


cos  9. 


=  £D-^  V3; 


FIG.  12.  Currents  and 
voltages  in  a  star-con- 
nected 3-phase  circuit, — 
radial  method  of  represen- 
tation. 


Total  power  =  \/^£D/S  cos  0. 


*(§iga).  If  the  neutral  point  of  the  supply  is  available,  measure  the 
voltage  between  it  and  O,  and  test  with  a  telephone  as  described  in  Appen- 
dix II.,  §44.  This  can  be  done  either  in  connection  with  the  present  test 
or  later  in  connection  with  Appendix  II. 


206  POLYPHASE   CURRENTS.  [Exp/ 

Since  line  voltage  is  ED  and  line  current  is  /s,  we  may  drop 
the  subscripts  and  write 

Total  power  =  V3  El  cos  6=  V3  El  X  power  factor, 

where  E  is  line  voltage  and  /  is  line  current.  This  is  the  custo- 
mary formula  for  power  in  any  balanced  3-phase  system,  no 
Line  Voltage  matter  how  connected.  In  the 

next  paragraph   it  will  be  de- 
rived for  a  delta-connection. 

§21  Delta-connection. — Con- 
nect the  same  three  equal*  re- 
sistances in  delta  to  a  3-phase 
supply,  as  in  Fig.  13.  Measure 
the  current  and  voltage  for  each 

7  resistance, — namely    the    delta 

F.O.  13.    Delta-  or  mesh-connection  of    ^^  j     ^  ^  ^^  (]jne) 
load  resistances. 

voltage  ED.     Also  measure  the 

line  current  /  and  the  star  voltage  Es,  if  the  neutral  O  of  the 
supply  system  is  accessible.     It  is  seen,  as  above,  ES  =  ED-+-  V3- 
Compare  the  measured  values  of  /  and  ID  with  the  expression 
(which  should  be  proved) 

/=  V3/D. 

Compute  the  power  for  each  resistance  ED!D,  and  compare 
with  the  power  found  for  the  same  resistances  in  star-connection. 

For  an  inductive  load,  we  should  multiply  by  cos  6  to  obtain 
the  true  power  in  each  resistance.  If  ED,  ID  and  6  are  the  same 
for  each  phase,  we  find  total  power  by  multiplying  by  3 ;  hence 

Total  power  =  3^0/0  cos  6. 
But 


hence 


*  Some  measurements  should  also  be  made  with  unequal  resistances. 


6-A]  GENERAL   STUDY.  207 


Total  power  =V3  £D/  cos  0, 
=  V3  El  cos  0, 
=  V3  -El  X  power  >  factor, 

where  E  and  /  are  line  voltage  and  line  current.  This  is  the 
customary  power  formula  for  any  balanced  3-phase  system,  as 
has  already  been  found  for  the  star-connection. 

§  22.  The  currents  and  voltages  for  the  delta-connection  can 
be  laid  off  by  the  radial  method  (see  Appendix  I.)  from  a  com- 
mon center,  giving  a  diagram  similar  to  Fig.  12. 

• 

Another  method  is  shown  in  Fig.  14,  in  which  the  voltages  are 
laid  off  as  a  triangle  (polygon  method)  and  the  currents  radially 
from  the  corners.  The  cur- 
rents  in  Fig.  14  are  drawn 
as  lagging.  These  currents 
are  /XY  (from  X  to  Y),  IYZ 
(from  Y  to  Z),  and  /zx  (from 
Z  to  X).  With  sign  reversed, 
the  latter  becomes  /xz,  meas- 
ured from  X  to  Z.  The  sum* 
of  /XY  and  /xz  gives  /.  If 
we  wish  to  select  signs  so  that 
the  sum  of  these  three  vectors 

is    zero,    we    must    reverse    the        FIG.  14.     Currents  and  voltages  in  a 

sign  of  I  so  as  to  give  the  line  ****<»»"«*   3-phase   circui.,-Poly- 

gon  or  mesh  method  of  representation. 

current  /'  ;  we  now  have  /',  /XY 

and  /xz  all  measured  from  X,  so  that  Law  (3)  of  Appendix  I. 

is  satisfied. 

§  23.  Transformer-connections  on  3-Phase  Circuits.  —  Trans- 
former secondaries  and  primaries  —  like  any  generating  or  re- 
ceiving circuits  —  can  be  connected  to  a  3-phase  circuit  by  A-,  Y-, 
T-  or  F-connections,  shown  in  Fig.  2. 

*(§22a).  The  current  /  is  the  sum  of  /xz  and  /XY  (both  measured 
from  X),  or  the  difference  between  /zx  and  /XY  (measured  one  towards 
and  the  other  away  from  X).  See  Laws  (3)  and  (4),  Appendix  I. 


208  POLYPHASE   CURRENTS.  [Exp. 

The  most  convenient  and  instructive  method  for  studying  the 
electrical  relations  of  these  connections  is  to  use  three  trans- 
formers with  the  same  ratio  of  transformation  (say  i:  i),  the 
primaries  and  secondaries  of  which  can  be  connected  in  any 
desired  manner. 

With  three  such  transformers  and  with  a  3-phase  supply  given, 
make  connections  in  the  following  six  ways : 

With  three  transformers: — 
—  (i)   Primaries  star-connected.     Secondaries  star-connected. 

(2)  "  "         "  "  delta 

(3)  "          delta     "  "  star 
.(4)           "             "         "                         "  delta 

With  two  transformers: — - 

(5)  Primaries  T-connected.     Secondaries  T-connected. 

(6)  "          V  V 

In  each  case  measure  all  electromotive  forces  and  construct 

electromotive     force     diagrams, 

_        v 

IT    comparing  computed  and  meas- 
ured results. 

The    star-    and    delta-connec- 
tions   have    already    been    dis- 
cussed; the  special  relations  of 
the   T-  and   F-connections,  will 
,  now  be  considered. 

*  ta  §  24.    T -connection. — For   the 

F.O.   ,5.     Relation  between  currents     r_connect;          ffleasure   the   yo,t. 
and  voltages   in  a   ./-connection. 

age  OZ,  Fig.  15,  and  note  that 

it  is  86.6  when  XY  is  100.  For  a  balanced  load,  the  three  cur- 
rents, /x,  /Y,  /z,  are  equal.  For  a  non-inductive*  load,  Fig.  15, 
the  current  in  transformer  XY  is  out  of  phase  with  the  elctro- 
motive  force  by  30°  and  the  power  factor  (cos  30°)  is  0.866; 

*  (§243).  For  an  inductive  load,  the  currents  take  the  positions  shown 
by  dotted  lines  in  Fig.  15;  7x  is  now  out  of  phase  more  than  30°,  and  /Y 
less  than  30°. 


6-A]  GENERAL   STUDY.  209 

in  transformer  OZ  the  current  is  in  phase  with  the  voltage, 
giving  unity  power  factor.  For  a  current  of  100  amperes,  on 
non-inductive  load,  we  have 

Volt- 
E  I  Power  Factor.         Watts.  amperes. 

Transformer  XY  100  100  0.866  8,666  10,000 

Transformer  OZ  86.6  100  i.oo  8,666  8,666 

17,333  18,666 

This  shows  that  the  power  output  of  each  transformer  is  the 
same ;  for  non-inductive  load  the  two  transformers  require  about 
8  per  cent,  more  transformer  capacity  (volt-amperes)  than  watts 
power  transmitted.  For  delta-  and  star-connection,  on  non- 
inductive  load,  no  excess  of  transformer  capacity  is  needed. 

The  T-connection  is  discussed  further  under  Polyphase  Trans- 
formation, Exp.  7- A,  where  it  is  shown  (§8)  that,  for  good 
regulation,  the  windings  OX  and  OF  on  one  transformer  must 
be  interspaced  so  as  to  reduce  the  magnetic  leakage  between 
them. 

The  neutral  point  in  a  T-connection  can  be  obtained  by  a  tap 
at  N  in  the  coil  OZ  (see  Fig.  9,  Exp.  7~A),  dividing  the  coil  into 
4  and  § . 

§  25.  V-connection  or  Open  Delta. — Draw  a  diagram  similar 
to  Fig.  15,  for  the  F-connection,  and  from  the  power  factor  of 
each  transformer  show  that  for  non-inductive*  load  this  connec- 
tion requires  15 J  per  cent,  more  transformer  capacity  than  power 
transmitted.  Obviously  a  F-connection  can  be  replaced  to  ad- 
vantage by  a  T-connection ;  even  using  the  same  two  transform- 
ers, there  will  be  an  advantage  in  the  change,  for  there  will 
be  less  voltage  on  one  of  the  transformers  and  hence  less  core 
loss. 

It  is  seen  that,  as  a  general  principle,  apparatus  in  which  cur- 
rents and  voltages  are  out  of  phase  require  greater  volt-ampere 

*  (§253).  Note  that  an  inductive  load  will  cause  the  power  factor  for 
one  transformer  to  become  more  and  for  the  other  less  than  cos  30°,  which 
will  make  the  regulation  better  on  one  and  worse  on  the  other. 
15 


210  POLYPHASE   CURRENTS.  [Exp. 

capacity  for  the  same  power  than  apparatus  in  which  currents 
and  voltages  are  in  phase. 

§  26.  A  Comparison. — In  comparing  the  relative  advantages  of 
transformer-connections,  it  is  to  be  borne  in  mind  that  three 
transformers  (even  though  of  somewhat  smaller  aggregate 
capacity)  will  usually  cost  more  than  two.  The  F-connection 
gives  the  least  voltage  per  transformer  and  the  least  insulation 
strain,  particularly  if  the  neutral  is  grounded;  for  this  reason  it 
is  to  be  preferred  on  high  potential  lines,  say,  20,000  volts  or 
over.  On  the  other  hand,  the  delta-connection  has  the  advan- 
tage that,  if  one  transformer  breaks  down,  the  remaining  two 
will  operate  F-connected;  at  moderate  voltages  (say,  under 
20,000  volts)  the  delta-connection  is  accordingly  to  be  preferred. 

In  the  delta-connection,  if  one  transformer  breaks  down,  each 
remaining  transformer  will  have  J  instead  of  ^  of  the  whole 
power  and  will  have  to  carry  the  line  current  instead  of  the 
delta  current.  By  what  per  centages  are  current  and  power  in 
each  transformer  thus  increased?  This  increase  would  cause 
abnormal  heating.  For  the  same  heating  (same  current)  show 
that  the  two  transformers  F-connected  will  carry  57!  per  cent, 
as  much  load  -as  the  three  delta-connected  transformers. 

With  transformers  delta-connected,  the  voltage  of  the  system 
can  be  increased  by  using  the  same  transformers  F-connected. 
In  a  new  system,  the  delta-connection  is  sometimes  installed  with 
a  view  to  changing  later  to  a  F-connection  and  a  higher  voltage. 

A  single  3-phase  transformer  requires  less  material  than  three 
single-phase  transformers  of  the  same  aggregate  capacity,  and 
is  more  efficient.  (See  Handbooks.)  The  three  single-phase 
transformers  may  be  cheaper  or  more  readily  obtained  because 
more  nearly  standard,  and  in  case  of  breakdown  one  third  and 
not  all  the  equipment  needs  be  replaced ;  in  other  respects  the 
single  3-phase  transformer  is  preferable  and  is  coming  more  and 
more  into  use. 

§  27.  SLv-Phase  Circuits. — A  6-phase  circuit  is  a  6-wire  cir- 


6-A]  GENERAL   STUDY.  211 

cuit,  the  potential  diagram  of  which  forms  a  hexagon.  Its  only 
use  is  in  connecting  transformer  secondaries  to  6-phase  syn- 
chronous converters.*  The  usual  and  best  method  for  obtaining 
a  6-phase  circuit  is  by  means  of  the  diametrical-connection,  as 
follows.  Three  transformers  have  primaries  connected  to  a 
3-phase  circuit.  The  six  wires  of  the  6-phase  circuit  may  be 
represented  by  the  apices  of  a  hexagon;  the  three  transformer 
secondaries,  Fig.  2  (a),  are  connected  so  as  to  form  diagonals 
or  diameters  of  the  hexagon.  The  three  neutral  or  middle  points 
of  the  secondaries  may,  or  may  not,  be  interconnected.  Connect 
transformers  in  this  manner,  with  the  neutrals  interconnected,  and 
test  with  a. voltmeter;  for  present  purposes  this  one  test  will  be 
sufficient. 

If  each  transformer  has  two  separate  secondaries  of  equal 
voltage,  these  six  coils  can  be  used  as  a  6-phase  supply  by  a  ring- 
or  mesh-connection  (each  coil  forming  diagrammatically  one  side 
of  a  hexagon)  ;  or,  a  6-phase  supply  can  be  obtained,  by  a 
double  T  or  double  delta,  one  T  or  delta  being  reversed  with 
respect  to  the  other.  A  double  F-connection  is  the  same  as  the 
diametral-connection.  One  advantage  of  the  diametral-con- 
nection is  that  it  gives  a  neutral  which  may  be  used  as  a  "  derived 
neutral "  for  a  3-wire  system  on  the  direct  current  service  from 
the  converter;  this  is  particularly  useful  in  lighting  systems. 

PART   IV. 

§  28.  Equivalent  Single-phase  Quantities. — Polyphase  quan- 
tities are  sometimes  reduced  to  equivalent  single-phase  values  for 

*  (§273).  A  3-phase  converter  may  be  increased  in  rating  40  or  50  per 
cent,  with  no  increased  losses  and  with  a  corresponding  higher  efficiency 
when  changed  to  6-phase  by  the  addition  of  three  more  collector  rings  and 
(if  necessary)  an  extension  of  the  commutator.  A  most  valuable  paper 
on  this  subject  is  one  by  Woodbridge  (A.  I.  E.  E.,  February  14,  1908),  who 
states  that  of  1,000,000  K.  W.  of  railway  converters,  one. third  are  6-phase; 
above  500  K.  W.  one  company  makes  all  converters  6-phase.  See  also 
Chap.  XI.,  Alternating  Current  Motors,  by  A.  S.  McAllister,  where  6-phase 
transformer  connections  are  given  in  detail. 


212  POLYPHASE    CURRENTS.  [Exp. 

simplicity  in  working  up  and  comparing  data  relating  to  poly- 
phase machinery. 

The  equivalent  single-phase  current  I'  (sometimes  called  total 
current)  in  any  balanced  polyphase  system  is  the  current  which, 
multiplied  by  the  line  voltage  and  power  factor,  gives  the  true 
(total)  power;  hence 

Total  power  —  El'  X  power  factor. 

For  a  2-phase  circuit,  the  equivalent  single-phase  current  /'  is 
evidently  twice  the  line  current. 

For  a  3-phase  circuit,  the  equivalent  single-phase  current  is 
V~3  times  the  line  current.  (In  a  delta-connection,  it  is  seen 
that  this  is  three  times  the  delta  current, — hence  the  significance 
of  total  current.) 

§  29.  Equivalent  single-phase  resistance  Rf  is  the  resistance 
which,  multiplied  by  the  square  of  the  equivalent  single-phase 
current,  gives  the  total  copper  loss  (=RT2).  It  will  be  found* 
that  for  star-  or  mesh-connection,  or  any  symmetrical  combination 
of  star  and  mesh, — 2-phase  as  well  as  3-phase, — R'  is  one  half  the 
resistance  measured  between  lines  of  one  phase. 

For  a  2-phase  circuit,  this  becomes  apparent  upon  inspection. 

For  a  3-phase  circuit,  with  the  three  equal  resistances  r  under 
test  connected  star  and  connected  delta,  determine  R'  and  /' ;  in 
each  case  compare  R  with  r  and  with  the  resistance  measured 
between  any  two  line-wires. 

Equivalent  single-phase  reactance  and  impedance  are  likewise 
one  half  the  measured  values  between  lines  of  one  phase. 

§  30.  Current  and  Voltage  per  Phase. — Current  per  phase  and 
voltage  per  phase  (or  phase  voltage)  are  more  commonly  used 
than  equivalent  single-phase  quantities ;  the  meaning  is  not  so 
definite,  but  can  generally  be  told  from  the  context.  The  terms 

*  See  Standard  Electrical  Handbook ;  or  Alternating  Current  Motors,  by 
A.  S.  McAllister,  in  which  equivalent  single-phase  quantities  are  exten- 
sively used. 


6-A]  GENERAL   STUDY.  213 

are  usually  so  used  that  the  total  power  in  a  2-phase  circuit  is 
twice  the  product  of  current  per  phase,  voltage  per  phase  and 
power  factor;  the  total  power  in  a  3-phase  circuit  is  three  times 
the  product  of  current  and  voltage  per  phase,  and  power  factor. 

In  a  2-phase  system,  there  is  little  chance  for  ambiguity. 

In  a  3-phase  system,  the  current  and  voltage  per  phase  (as 
denned  above)  may  be  either  the  star  (line)  current  and  star 
voltage,  or  the  delta  current  and  delta  (line)  voltage.  In  either 
case,  the  total  power  is  three  times  the  power  per  phase.  Using 
line  current,  we  must  use  star  voltage ;  using  line  voltage,  we 
must  use  delta  current.  It  will  be  remembered  that,  if  line  cur- 
rent and  line  voltage  are  used,  the  total  power  is  \/3  times  their 
product  multiplied  by  power  factor. 


APPENDIX  I. 

VECTOR  ADDITION  OF  ALTERNATING  CURRENTS  AND  ELECTRO- 
MOTIVE FORCES  IN  A  NETWORK  OF  CONDUCTORS. 

§31.  Laws  of  Vector  Addition  and  Subtraction. — Any  hill  may  be 
considered  to  be  up  or  down  according  to  the  direction  in  which  one 
is  walking;  the  difference  in  level  may  be  considered  positive  or 
negative.  In  the  same  way  difference  of  potential  may  be  considered 
as  positive  or  negative  according  to  the  sense  in  which  it  is  taken — 
that  is,  according  to  the  direction  one  takes  in  proceeding  around  a 
circuit  or  from  point  to  point  in  a  circuit. 

Consider  a  network  of  highways  in  a  hilly  country.  If  from  any 
starting  point  one  proceeds  by  any  route  or  circuit  back  to  the  starting 
point,  he  will  find  himself  at  the  original  level — the  plus  hills  and  the 
minus  hills  adding  up  to  zero.  On  different  trips  he  may  traverse 
the  same  hill  in  opposite  directions,  giving  it  one  time  a  plus  and  the 
other  time  a  minus  sign.  This  would  be  true  at  any  instant,  even  if 
the  surface  were  rising  and  falling,  as  in  an  imaginary  earthquake  or 
on  the  surface  of  the  ocean. 

Consider  now  a  network  of  conductors.  If  from  any  starting  point 
one  proceeds  by  any  route  or  circuit  back  to  the  starting  point,  he  will 


214  POLYPHASE    CURRENTS.  [Exp. 

reach  the  original  potential ;  the  algebraic  sum  of  the  potential  differ- 
ences at  any  instant,  taken  in  the  proper  sense,  adding  up  to  zero. 

For  an  alternating  current  circuit  in  which  currents  and  potential 
differences  vary  harmonically  and  can  be  represented  by  vectors, 
algebraic  addition  is  used  for  instantaneous  values  and  vector  addi- 
tion for  maximum  or  for  effective  values;  hence,  for  maximum  or 
effective  values  we  have  the  modified  statement  of  Kirchhoff's  Law: 

§32.  Law  (/).  Vector  Addition  of  Electromotive  Forces:  Gen- 
eral Law. — In  proceeding  completely  around  any  mesh  or  number  of 
meshes  in  an  alternating  current  system  of  conductors,  the  vector 
sum  of  all  the  differences  in  potential  is  zero;  such  vectors  form  a 
closed  polygon. 

For  this  vector  addition,  electromotive  forces  are  represented  by 
arrows,  the  tip  of  one  to  the  feather  of  the  next,  which  must  be  in 
sequence  according  to  the  direction  in  which  we  proceed  around  the 
circuit.  A  coil  xy  may  have  an  electromotive  force  represented  by  a 
vector  XY,  as  measured  from  x  to  y.  Taken  in  the  opposite  sense 
(by  traversing  the  circuit  in  the  opposite  direction)  the  electromotive 
force  would  be  YX,  the  same  vector  with  arrow  reversed. 

To  illustrate*  further  this  addition,  from  a  point  O  on  the  side  of 
a  hill,  let  two  paths  ascend:  one  to  the  point  A  (elevation  100)  ;  the 
other  to  B  (elevation  90).  If  a  man  starts  at  A,  descends  to  0, 
ascends  to  B  and  back  to  Af  the  ascents  and  descents  add  to  zero 
(—100;  +90;  +  10). 

To  illustrate  the  special  case  of  subtraction,  if  the  sense  or  sign  of 
one  quantity  be  reversed:  let  two  men  start  from  O,  one  ascending 
to  A  (-{-  100)  and  the  other  to  B  (+90).  The  difference  in  their 
level  is  now  the  difference  between  -f-  100  and  +  90,  which  illustrates 
the  following  law : 

§33.  Law  (<?).  Vector  Subtraction  of  Electromotive  Forces: 
Special  Law. — In  an  alternating  current  system,  if  two  electromotive 
forces  are  separately  measured  away  from  a  common  point  (as  OA 
and  OB}  the  difference  in  potential  between  their  outer  ends  (A  and 
B}  will  be  the  vector  difference  of  the  two  electromotive  forces  (OA 
and  OB). 

*  For  unvarying  potentials  or  instantaneous  values  of  varying  poten- 
tials this  is  a  correct  analogy ;  for  the  vector  addition  of  varying  quantities 
it  is  merely  an  illustration. 


6-A]  GENERAL   STUDY. 


2  I 


The  discussion  of  Figs.  3,  4  and  5  illustrates  the  application  of  Laws 
(i)  and  (2). 

The  modified  form  of  KirchhofFs  Law  for  current  becomes: 

§34.  Law  (j).  Vector  Addition  of  Currents:  General  Law. — At 
any  point  in  an  alternating  current  system  the  vector  sum  of  the 
currents  measured  all  towards  or  all  away  from  that  point  is  zero; 
such  vectors  form  a  closed  polygon. 

§  35.  Law  (4}.  Vector  Subtraction  of  Currents:  Special  Law. — At 
any  point  in  an  alternating  current  system  where  three  currents  come 
together,  if  one  current  is  measured  towards  and  the  second  away 
from  that  point,  the  third  current  will  be  the  vector  difference  of 
the  two. 

The  discussion  of  Fig.  14  illustrates  the  application  of  Laws  (3) 
and  (4). 

§  36.  Notation. — There  is  no  universally  adopted  notation  for  poly- 
phase circuits.  The  most  complete  and  least  ambiguous  method  is 
to  letter  every  junction  or  point  on  the  diagram  of  connections  and 
to  use  two  letters  (as  subscript  if  desired)  in  the  proper  sequence 
to  designate  the  vector  current  or  electromotive  force  between  two 
points.  Thus,  from  X  to  Y  we  may  have  electromotive  forces  XY 
or  £XY;  in  the  reverse  sense,  YX  or  £YX  ;  similarly,  we  may  speak 
of  the  currents  XY  or  /XY  and  YX  or  /YX.  This  makes  definite  the 
direction  or  sign  of  the  vector  quantity  in  every  case.  In  some  cases, 
particularly  the  simpler  ones,  the  complete  definiteness  is  not  needed 
(being  unessential  or  obvious)  and  a  single  subscript  is  then  simpler, 
as  ED,  Es,  I  A,  IB.  In  general  the  double-subscript  notation  is  to  be 
recommended  on  account  of  its  exactness,  as  illustrated  in  the  dis- 
cussion of  Fig.  14. 

§37.  In  applying  Law  (i)  it  is  necessary,  in  order  to  obtain  a 
vector  sum  of  zero  in  proceeding  from  a  generator  around  a  circuit 
and  back  to  the  generator,  to  take  the  generated  electromotive  forces 
or  counter  electromotive  forces  in  each  part  of  the  circuit:  thus,  the 
electromotive  force  produced  by  self-induction  90°  behind  the  cur- 
rent (not  that  to  overcome  self-induction  90°  ahead  of  the  current)  ; 
and  the  electromotive  force  produced  by  resistance,  in  direction 
exactly  opposite  to  the  current  (not  the  electromotive  force  to  over- 
come resistance  which  is  in  phase  with  current).  This  becomes 
obvious  upon  inspection  of  the  triangle  for  the  electromotive  forces 


216  POLYPHASE   CURRENTS.  [Exp. 

in  a  simple  circuit,  the  hypotenuse  of  which  is  E,  one  side  RI  and 
the  other  side  XI;  the  principle  is  applied  in  the  discussion  of  Fig.  10. 

§  38.  Polygon  or  Mesh  Method  of  Representation. — As  applied  to 
electromotive  forces,  there  is  in  this  method  of  representation  a  cer- 
tain similarity  between  the  diagram  of  connections  and  the  diagram 
for  electromotive  forces.  It  seems  a  natural  method  to  apply  in 
many  cases,  as  in  Figs.  8,  9,  10.  There  is  no  essential  difference 
between  it  and  the  topographic  method.  Law  (i),  above,  applies 
directly  and  the  electromotive  forces  around  any  mesh  have  a  vector 
sum  of  zero,  introducing  arrows  with  feather  to  tip  in  sequence. 
(Compare  analogy  of  network  of  highways,  §31.) 

As  applied  to  currents,  the  three  currents  drawn  radially  in  Fig. 
12  may  be  drawn  as  a  closed  polygon.  So  also  in  Fig.  7.  Compare 
likewise  Fig.  14. 

§  39.  Radial  Method  of  Representation. — In  this  method  all  vectors 
for  currents  and  electromotive  forces  are  drawn  radially  from  a 
common  center.  This  method  is  advocated  by  some  for  all  cases 
(Porter,  Electric  Journal,  September,  1907),  together  with  the  double 
subscript  notation,  in  order  that  in  involved  problems  ambiguity  can 
be  minimized.  For  a  star-connection  the  application  is  obvious. 
For  a  delta-connection,  we  have  the  same  radial  diagram  as  for  the 
star-connection.  See  Fig.  12. 

A  modified  radial  method,  with  vectors  from  several  centers,  is 
illustrated  in  Figs.  14  and  15,  and  for  particular  cases,  as  in  those 
illustrated,  possesses  some  advantages. 

§  40.  Preferred  Method. — It  is  not  proposed  to  advocate  here  a  par- 
ticular convention  but  rather  to  assist  in  making  underlying  principles 
clear.  One  may  choose  or  develop  one  method  and  apply  it  in  all 
cases;  or  he  may  select  the  method  which  is  simplest  or  clearest  for 
each  particular  case.  The  important  point  is  to  see  clearly  the  sig- 
nificance of  whatever  method  is  used. 


6-AJ  GENERAL   STUDY.  21? 

APPENDIX  II. 
TRIPLE   HARMONIC   IN  DELTA  AND   STAR   CONNECTION. 

§  41.  In  a  circuit  supplying  current  to  a  transformer,  induction 
motor  or  similar  apparatus  with  iron,  hysteresis  in  the  iron  intro- 
duces* in  the  exciting  current  odd  harmonics  of  3,  5,  7,  9,  etc.,  times 
the  fundamental  frequency. 

In  a  3-phase  system,  if  three  transformers  have  their  primaries 
either  star-  or  delta-connected,  the  currents  in  the  three  transformers 
will  have  a  phase  difference  of  one  third  of  the  fundamental  period. 
The  third  harmonic  due  to  hysteresis  will  accordingly  have  the  same 
phase  in  each  of  the  three  transformers.  This  will  be  seen  by 
sketching  curves  for  the  fundamental  and  third  harmonic,  and  shift- 
ing the  curves  to  left  or  right  one  third  of  the  fundamental  period, 
which  is  one  full  period  for  the  third  harmonic.  In  a  3-phase  system 
all  harmonics  divisible  by  3,  as  the  9th,  I5th,  etc.,  will  likewise  have 
the  same  phase  in  each  transformer. 

For  a  5-  or  7-phase  system,  the  harmonics  thus  appearing  would 
be  5,  15,  25  and  7,  21,  35,  etc.,  respectively. 

In  an  even  phase  (single-  or  2-phase)  system,  even  harmonics  only 
could  appear;  but  no  even  harmonics  are  produced  by  hysteresis. 

These  facts  can  be  shown  by  curves  taken  by  the  method  of  instan- 
taneous contact  or  the  oscillograph.  A  set  of  such  curves  has  been 
published  and  discussed  by  E.  J.  Berg  (Electrical  Energy,  p.  154). 

§  42.  Thirdf  Harmonic  in  Delta-connection.  —  If  the  transformer 
primaries  are  delta-connected,  the  harmonics  due  to  hysteresis  for 
the  three  transformers  are  in  phase  and  form  a  current  which  circu- 
lates around  the  delta  but  does  not  appear  in  the  line.  The  delta 
current  ID  may  accordingly  be  5  or  10,  per  cent,  more  than  the  line 
current  I,  divided  by  \/3-  If  H  is  the  current  (third  and  higher  har- 
monics) caused  by  hysteresis,  we  have:}:  /D  — 


*  Compare  "  The  Effect  of  Iron  in  Distorting  Alternating  Current  Wave 
Form,"  A.  I.  E.  E.,  September,  1906,  and  its  discussion  by  Steinmetz. 

fThe  third  harmonic  is  mentioned,  being  most  important;  it  will  be 
understood  that  the  ninth,  fifteenth,  etc.,  are  included  when  only  the  third 
is  mentioned. 

$  (§  423).  If  A  and  B  are  currents  or  voltages  of  any  two  frequencies, 
the  total  effective  value  is  \fAz-\-B2.  This  is  easily  shown  experimentally 


21  8  POLYPHASE    CURRENTS.  [Exp. 

Suppose,   for  example,  /  is   173   and  ID  is   105   instead  of   100    (an 


increase  of  5  per  cent.);  then  H  =  ^/io$2  —  ioo2  =  32. 

In  the  laboratory  measure  /  and  ID  and  calculate  H.  Compute  H 
as  per  cent,  of  /  -f-  \/3  J  also  compute  the  per  cent,  increase  in  /D  over 
/-f-\/3-  Although  noticeable  at  no  load,  the  percentage  difference 
practically  disappears  under  load,  for  H  remains  constant  and  hence 
is  relatively  smaller  when  /  and  ID  become  large. 

§43.  Third  Harmonic  in  Star-connection.  —  If  the  transformer  pri- 
maries are  y-connected,  the  third  harmonic  caused  by  hysteresis 
will  be  in  the  same  phase  in  the  three  transformers  and  will  tend  to 
flow  to  or  from  the  neutral  simultaneously  in  the  three.  The  star 
voltage  Es  will  thus  be  more  than  the  line  voltage  E  divided  by  \/3, 


thus  E$  =  V(£n-  V3)2  +  £n2,  where  EH  is  voltage  due  to  hystere- 
sis. If  the  neutral  is  insulated  no  current  due  to  these  harmonics 
can  flow.  If  there  is  a  return  circuit  from  the  neutral,  through 
ground  or  a  fourth  wire,  a  current  of  triple  frequency  will  flow;  but 
no  current  of  fundamental  frequency  will  flow  in  the  neutral  if  the 
line  voltages  are  symmetrical. 

§  44.  The  third  harmonic  in  the  neutral  can  be  prettily  shown  in 
the  laboratory  by  means  of  a  telephone,  which  should  be  protected 
by  a  resistance  in  series,  or  in  shunt,  or  both,  or  by  connecting  through 
a  transformer.  Connect  to  a  3-phase  supply  three  transformers  or 
other  coils*  with  iron;  the  more  nearly  similar  these  are  the  better. 
Let  O'  be  the  neutral  of  the  three  coils  and  let  O  be  the  neutral  of 
the  supply  system.  (If  the  supply  system  has  no  neutral,  one  may  be 
obtained  by  three  F-connected  resistances.)  Connect  the  telephone 
between  0  and  O'.  If  the  coils  are  well  balanced,  the  fundamental 
will  be  perhaps  scarcely  discernible;  the  third  harmonic  will  sound 
very  clearly  an  octave  and  a  fifth  (do  to  sol}  above  the  fundamental; 
the  ninth,  if  discernible,  is  the  same  interval  above  the  third.  On 
a  64-cycle  circuit,  the  fundamental  is  C  with  harmonics  g,  d",  b",  etc. 

If  there  is  any  question  as  to  what  is  the  fundamental,  it  can 
usually  be  told  by  listening  to  various  apparatus  in  the  laboratory; 
or  by  connecting  the  telephone,  with  a  series  resistance,  to  the  supply 
circuit. 


by  measuring  the  total  and  separate  voltages  when  two  sources  are  put 
in  series.     Do  not  short  circuit  one  source  on  the  other. 

*  Shunt  choking-coils  used  for  series  lighting  are  suitable  for  this. 


6-A]  GENERAL   STUDY.  219 

If,  instead  of  coils  with  iron,  three  resistances  are  used,  the  har- 
monics cannot  be  heard;  the  fundamental  will  no  doubt  be  heard 
due  to  lack  of  perfect  symmetry,  and  will  become  louder  if  the  circuits 
are  thrown  more  out  of  balance. 

§  45.  If  an  electrostatic  voltmeter  is  available,  connect  it  between 
O  and  0'  (in  place  of  the  telephone)  and  measure  the  hysteresis 
voltage,  EH.  Measure  also  from  one  line  wire  X,  the  voltages 
OX  and  O'X,  the  latter  being  the  larger  on  account  of  hysteresis 
harmonics. 

Compute  O'X  from  the  formula  O'X  =  \J~(OX~Y "+~E?  and  com- 
pare with  the  measured  value.  It  is  to  be  noted  that  OX  =  E-+-  V3 
and  O'X  =  Es.  By  what  per  cent,  is  Es  greater  than  E  -=•-  >/3  ? 
What  per  cent,  is  EH  of  E-i-  V3? 

§  46.  With  a  voltmeter*  measure  the  line  voltage  XYZ  and  con- 
struct a  triangle  as  in  Fig.  16.  Measure  also  the  three  star  voltages 
O'X,  O'Y,  O'Z  and  lay  them  off  as  shown,  each  one  twice.  A  supply 
neutral  is  not  necessary  for  this  test.  Cut 
out  the  diagram  on  the  heavy  lines  and 
fold  on  the  light  lines,  bringing  the  three 
points  0'  together  so  as  to  form  a  pyramid. 
The  height  of  the  pyramid  represents  the 
voltage  EH  due  to  hysteresis  harmonics. 

§  47.  The  foregoing  illustrates  the  fact 
that  vectors  in  a  plane  can  exactly  represent 
only  currents  and  electromotive  forces 
which  are  simple  sine  functions;  the  error 
due  to  harmonics  is  commonly  neglected. 

§  48.  Generator  Coils. — If  there  is  a  third  harmonic  in  the  generated 
electromotive  force,  with  the  generator  coils  delta-connected  it  cannot 
appear  in  the  line  but  will  appear  as  a  circulating  current  in  the  delta. 
This  may  cause  appreciable  heating  if  the  harmonic  is  large. 

§49.  The  third  harmonic  can  appear  on  the  line  only  in  case  the 
generator  coils  are  F-connected  and  have  the  neutral  connected  to 
ground  or  a  4th  wire.  If  the  line  is  not  grounded  also  at  the  receiv- 

*  Use  an  electrostatic  voltmeter ;  although  this  is  not  important  with 
large  transformers,  it  becomes  necessary  in  case  the  coils  or  transformers 
are  small,  as  the  current  taken  by  an  ordinary  voltmeter  may  cause  con- 
siderable error. 


220  POLYPHASE    CURRENTS.  [Exp. 

ing  end  or  a  4th  wire  return  used,  the  potential  of  the  line  as  a  whole 
will  be  raised  by  this  electromotive  force  of  triple  frequency. 

APPENDIX  III. 
COPPER  ECONOMY  OF  VARIOUS  SYSTEMS. 

§  50.  In  figuring  copper  economy,  it  is  to  be  assumed  that  all  sys- 
tems compared  are  to  have  the  same  line  loss  and  per  cent,  resistance 
drop. 

As  a  general  principle,  in  any  given  system,  the  amount  of  copper 
necessary  varies  inversely  as  the  square  of  the  voltage;  thus,  if  the 
voltage  is  doubled,  the  current  will  be  halved  and  the  copper  reduced 
to  one  fourth,  increasing  R  four-fold.  This  gives  the  same  RI2  loss 
in  the  line  and  the  same  per  cent.  RI  drop. 

Any  comparison  of  systems  should,  therefore,  be  made  on  the  basis 
of  equal  voltage;  this  may  mean  either  the  greatest  voltage  between 
any  two  line  wires  or  the  voltage  between  any  wire  and  the  neutral. 
This  latter  becomes  more  significant  when  the  neutral  is  grounded. 

§  51.  On  the  Basis  of  the  Same  Voltage  E$  from  the  Line  Wire  to 
Neutral. — On  this  basis  all  symmetrical  alternating  systems  give  the 
same  copper  economy,  as  will  be  seen  from  the  following.  Let  us 
consider  all  wires  to  be  of  a  given  size  and  to  carry  a  given  current  I, 
thus  giving  the  same  drop  and  loss  per  wire.  We  then  have 

Single-phase,         2  wires:  amount  of  copper  2;  power  =  2  E$I. 

Three-phase,          3  wires :  amount  of  copper  3 ;  power  =  3 

Quarter-phase,       4  wires :  amount  of  copper  4 ;  power  =  4 

w-phase,  n  wires  :  amount  of  copper  n ;  power  =  « 

The  amount  of  power  is  seen  to  be  proportional  to  the  amount  of 
copper,  giving  therefore  equal  copper  economy  for  all  systems  on  the 
basis  of  equal  voltage  between  the  line  and  the  neutral  or  ground. 

§  52.  On  the  Basis  of  the  Same  Voltage  Between  Line  Wires. — 
Between  line  wires  the  voltage  is  2£s  for  the  single-phase  (or 
quarter-phase)  system  and  V3-Es  for  the  3-phase  system.  To  make 
the  voltage  between  line  wires  equal  in  these  systems,  the  voltage  in 
the  3-phase  system  can  be  increased  in  the  ratio  \/3 :  2.  The  amount 
of  copper  can  accordingly  be  reduced  (see  §  50)  inversely  as  the 
square  of  this  ratio,  namely,  4:3.  Hence,  for  the  same  line  voltage, 
a  3-phase  system  requires  75  per  cent,  as  much  copper  as  a  single- 
phase  or  quarter-phase  system. 


6-A]  GENERAL   STUDY.  221 

§  53.  Direct  Currrent  System. — A  direct  current  system  has  the 
same  copper  economy  as  a  single-phase  system,  when  the  direct  cur- 
rent voltage  is  made  equal  to  the  effective  (sq.  rt.  of  mean  sq.)  value 
of  the  alternating  voltage. 

If,  however,  the  direct  current  voltage  is  increased  so  as  to  equal 
the  maximum  value  of  the  alternating  current  voltage,  the  direct 
current  voltage  is  increased  in  the  ratio  of  I : \/2  and  the  copper  is 
decreased  as  the  inverse  square  of  this  ratio.  The  direct  current 
system  then  requires  only  one  half  the  copper  of  a  single-phase  or 
two  thirds  the  copper  of  a  3-phase  system,  on  the  basis  of  equal  volt- 
age between  wires. 

A  direct  current  system  would,  therefore,  be  more  economical  of 
copper  than  any  other  system,  at  the  same  voltage. 

§  54.  Choice  of  Systems. — On  account  of  copper  economy  and  the 
simplicity  due  to  the  use  of  only  two  wires,  direct  current  would  be 
superior  to  any  alternating  current  system,  if  it  were  not  for  lack 
of  simple  and  suitable  means  for  transforming  direct  current  so  as 
to  obtain  the  advantage  of  high  potential  transmission  with  low 
potential  generation  and  utilization.  In  the  case  of  alternating  cur- 
rents, these  means  are  provided  for  by  the  transformer  which  makes 
alternating  current  systems  so  flexible  that  they  are  practically 
always*  used  for  long  distance  transmission,  instead  of  direct  current. 

In  comparing  alternating  current  transmission  systems,  the  choice 
is  to  be  made  between  single-phase — with  its  simpler  line  construction, 
fewer  insulators,  etc. — and  3-phase,  requiring  only  75  per  cent,  as 
much  copper.  If  these  were  all  the  factors,  single-phase  transmission 
systems  would  be  more  common  than  they  are,  the  simplicity  offsetting 
the  poorer  copper  economy.  An  important  and  perhaps  a  determin- 
ing factor,  however,  is  the  superiority  of  polyphase  as  compared  with 
single-phase  machinery  (§2)  ;  for  this  reason  a  polyphase  system  is 
commonly  preferred,  quite  aside  from  considerations  of  copper  econ- 
omy. Of  polyphase  systems,  the  3-phase  system  is  most  economical 
and  is  therefore  the  system  in  general  use. 

*  (§  54a)-  In  a  f£W  cases  high  potential  direct  current  has  been  used  for 
power  transmission,  notably  in  the  Thury  system.  This  is  essentially  a 
constant  current  system.  The  high  potential  is  obtained  by  generators  in 
series ;  the  motors  are  likewise  in  series.  See  Land.  Electrician,  March 
19,  1897;  New  York  Elect.  Rev.,  January,  1901. 


222  POLYPHASE   CURRENTS.  [Exp. 

EXPERIMENT  6-B.  Measurement  of  Power  and  Power  Fac- 
tor in  Polyphase  Circuits. 

PART    I.      GENERAL    DISCUSSION.         . 

§  i.  Preliminary. — For  measuring  power  in  any  3-wire  sys- 
tem, the  best  method  is  the  two-wattmeter  method  §  23 ;  for  the 
particular  case  of  a  balanced  3-phase  load,  some  one-wattmeter 
method,  §§  32-9,  may  be  used. 

For  measuring  power  in  systems  with  more  than  three  wires, 
the  n — i  wattmeter  method  of  §  16  is  correct  for  all  cases;  for 
the  particular  case  of  a  balanced  2-phase  load,  on  a  4-wire  sys- 
tem, the  method  of  §  10,  employing  two  wattmeters,  may  be  used. 

An  unknown  load  should  not  be  assumed  to  be  balanced. 

It  will  be  understood  that,  in  cases  where  several  wattmeters 
are  described  as  being  required,  a  single  instrument  may  be 
used  and  shifted  by  suitable  switches  from  circuit  to  circuit, 
readings  being  taken  successively  in  the  different  positions. 

§2.  Separate  Phase  Loads. — In  any  single-phase  system 
power  is  measured  by  means  of  a  wattmeter,  the  current  coil 
being  connected  in  series  and  the  potential  coil  in  parallel  with 
the  circuit,  as  discussed  in  Appendix  III.,  Exp.  5-A.  An  exten- 
sion of  this  method  can  be  applied  to  a  polyphase  system,  if  the 
phases  are  separately  accessible  so  that  the  load  of  each  phase 
can  be  separately  measured.  A  wattmeter  is  then  used  for  each 
phase  load,  with  current  coil  in  series  and  potential  coil  in  parallel 
with  the  particular  load  being  measured,  the  total  power  being 
the  arithmetical  sum  of  the  several  wattmeter  readings. 

For  example,  to  measure  the  power  in  three  star-connected  re- 
sistances on  a  3-phase  circuit  by  this  method,  three  wattmeters 
would  be  required,  each  current  coil  carrying  the  star  (or  line) 
current  and  each  potential  coil  being  subjected  to  the  star  voltage. 
With  three  resistances  delta  connected,  three  wattmeters  would 


6-B]  MEASUREMENT   OF    POWER.  223 

also  be  required,  each  current  coil  carrying  the  delta  current  and 
each  potential  coil  being  subjected  to  the  delta  (or  line)  voltage. 

§  3.  This  method  of  measuring  the  separate  phase  loads  is 
simple  in  principle  and  is  commonly  used  on  a  2-phase  circuit 
(§6),  but  it  is  not  capable  of  general  application  inasmuch  as 
phase  loads  are  not  always  separable.  On  a  3-phase  circuit — 
in  testing,  for  example,  a  3-phase  induction  motor — it  may  be 
impossible  to  measure  delta  current  or  star  voltage,  so  that  some 
method  not  requiring  either  of  these  measurements  becomes 
necessary;  furthermore,  the  method  is  open  to  objection  on 
account  of  the  number  of  measurements  required, — unless  the 
assumption  is  made  that  all  phases  are  alike,  so  that  measure- 
ments are  necessary  on  one  phase  only. 

§4.  Polyphase  Power  Factor. — A  polyphase  system  is  a  com- 
bination of  single-phase  elements.  If  E,  I  and  W  are,  respectively, 
the  voltage,  current  and  power  for  any  separate  element,  the 
power  factor  for  that  element  is  W  -~EI,  by  definition.  When 
the  separate  elements  or  phases  of  a  polyphase  system  have  the 
same  power  factor,  this  is  the  power  factor  for  the  whole  system. 

§5.  When,  however,  the  separate  elements  have  different  power 
factors,  there  is  no  one  power  factor  that  has  a  definite  value  or 
physical  significance  for  the  whole  system.  It  is  convenient, 
however,  to  obtain  a  kind  of  average  power  factor  for  the  system, 
the  value  of  which  will  depend  upon  the  method  used  in  its 
determination.*  An  average  power  factor  may  be  satisfactorily 
determined  when  the  separate  phases  are  nearly  alike,  but  has 
little  meaning  when  they  are  widely  different. 

§6.  Two-phase  Load. — Two-phase  power  is  usually  measured 
by  two  wattmeters,  one  on  each  phase,  as  just  described. 

§7.  When  the  phases  are  independent,  as  in  a,  Fig.  I,  Exp. 
6-A,  the  measurements  differ  in  no  respect  from  measurements 
made  on  single-phase  circuits. 

*(§5a).  See  A.  S.  McAlliser,  Alternating  Current  Motors,  p.  12;  A. 
Burt,  Three-phase  Power  Factor,  A.  I.  E.  E.,  p.  613,  Vol.  XXVIL,  1908. 


224  POLYPHASE   CURRENTS.  [Exp. 

§8.  On  a  3-wire,  2-phase  circuit,  as  in  b,  Fig.  i,  Exp.  6-A,  the 
same  method  may  also  be  used,  the  two  wattmeter  current  coils 
being  located  in  the  two  "  outer  "  conductors,  A  and  B,  respectively. 
With  the  wattmeters  thus  located,  the  sum  of  their  two  readings 
will  give  the  true  power  (§23)  for  any  load  whatsoever,  even 
when  part  of  the  load  is  between  A  and  B.  (These  connections 
are  seen  in  Fig.  I,  in  which  X  and  Y  are  the  outer  conductors 
and  Z  is  the  common  conductor  or  return.) 

§  9.  When  the  load  in  a  3-wire  2-phase  system  is  balanced  and 
there  is  no  load  between  the  two  outer  conductors  A  and  B,  one 
wattmeter  may  ;be  conveniently  used  by  connecting  the  current 
coil  in  the  common  conductor;  one  end  of  the  potential  coil  is 
connected  to  the  common  conductor  and  the  other  end  connected 
first  to  one  and  then  to  the  other  outer  conductor.  A  reading* 
is  taken  in  each  position  and  the  algebraicf  sum  gives  the  total 
power.  (The  connections  are  seen  in  Fig.  7,  in  which  Z  is  the 
common  conductor.)  A  3-wire  2-phase  circuit  is  likely  not  to 
be  balanced  (§  14,  Exp.  6-A)  and  the  method  should  be  used 
with  caution. 

§  10.  On  a  4-wire,  quarter-phase,  2-phase  system,  as  in  c  and 
d,  Fig.  i,  Exp.  6-A,  two  wattmeters,  one  on  each  phase,  will  give 
the  correct  power  only  when  the  load  is  balanced.  The  method 
may  be  used  for  testing  a  single  machine,  but  not  for  measuring 
the  power  of  a  circuit  when  the  character  of  its  load  is  unknown. 

*  (§9a).  For  a  balanced  load,  power  can  be  determined  from  a  single 
reading  of  the  wattmeter  by  connecting  the  current  coil  in  the  common 
conductor  and  connecting  the  potential  circuit  from  the  common  conductor 
to  the  middle  point  of  two  approximately  equal  non-inductive  resistances, 
Ri  Rz,  connected  across  the  two  outer  conductors  as  in  Fig.  5.  A  single 
reading  of  the  wattmeter  gives  one  half  the  total  power,  if  the  wattmeter 
is  calibrated  as  a  single-phase  instrument  with  Rt  and  R2  connected  in 
parallel  with  each  other  and  in  series  with  the  potential  circuit  (§36a). 
See  also  §  33a. 

t  (§Qb).  For  low  power  factors,  when  0  exceeds  45°,  the  reading  of  the 
wattmeter  in  one  position  is  negative.  The  similar  case  for  a  3-phase 
circuit  is  fully  discussed  later. 


6-B]  MEASUREMENT   OF    POWER.  225 

That  the  method  is  not  generally  correct  will  be  seen  by  assum- 
ing the  current  coils  of  the  two  wattmeters  to  be  connected  in 
two  of  the  lines,  as  A  and  B ;  neither  wattmeter  would  then 
record  a  single-phase  load  drawing  current  from  the  other  two 
lines,  A'B'. 

On  a  4-wire  system,  with  unbalanced  load,  at  least  three  watt- 
meters must  be  used,  §  16. 

§  ii.  Po^ver  Factor  in  a  Tzvo-phase  Circuit. — If  E,  I  and  W 
are  measured  on  one  phase  of  a  2-phase  circuit,  W  -f-  El  is  the 
power  factor  for  that  phase,  §  4.  This  may  be  called  the  cosine 
method  for  determining  power  factor,  since  W  -7-  El  =  cos  6 
when  currents  and  electromotive  forces  are  represented  by  sine 
waves. 

§  12.  The  following  tangent  method  for  determining  power 
factor  from  two  readings  of  the  wattmeter  will  be  found  simple 
and  often  convenient. 

The  current  coil  of  the  wattmeter  is  connected  in  one  line  of 
phase  A  ;  the  potential  coil  is  connected  across  phase  A,  whose 
voltage  is  EA.  The  wattmeter  now  reads  the  power  volt-amperes 
or  true  watts 

(1)  Wi  =  EAIAcosO. 

Transfer  the  potential  coil  to  phase  B,  whose  voltage  is  E-B. 
The  wattmeter  now  reads  the  wattless  or  quadrature  volt- 
amperes  (sometimes  called  wattless,  or  quadrature,  watts), 

(2)  ^2  =  £B/Asin0. 
Dividing  the  second  reading  by  the  first, 

(3)          5= a— 

Tan  0,  and  hence  power   factor    (cos0),   is   determined  by  the 
ratio  of  the  two  readings.     Usually  EB  =  £A,  so  that  tan  0  = 
W2  _r_  w±.      The  power   factor  thus   determined   is   the  power 
16 


226  POLYPHASE   CURRENTS.  [Exp. 

factor  of  phase  A  ;  6  is  the  phase  difference  between  JA  and  E&. 
The  method  assumes  that  EA  and  EE  differ  90°  in  phase  and  that 
electromotive  forces  and  currents  follow  a  sine  law. 

The  advantage  of  the  tangent  method  is  its  simplicity  and  inde- 
pendence of  the  calibration  of  instruments.  The  method  can  be 
used  for  determining  the  power  factor  of  a  single-phase  load, 
drawn  from  a  2-phase  supply,  and  a  somewhat  similar  method 
can  be  used  for  determining  the  power  factor  of  a  3-phase  load, 
§§28,  38,  41. 

§  13.  The  value  of  6  and  power  factor  can  be  found  by  the 
sine  method  directly  from  (2);  thus,  sin  0  =  W2  -f-  £B/A.  For 
a  single-phase  or  2-phase  load  there  is  little  advantage  in  this 
method,  which  is  useful,  however,  on  3-phase  circuits,  §  43. 

§  14.  The  "  cosine "  method  gives  correct  power  factor  by 
definition  and  is  general,  being  independent  of  wave  form.  The 
"  tangent "  and  "  sine "  methods  are  based  on  the  assumption 
that  voltages  and  currents  follow  a  sine*  law.  The  "  cosine  " 
and  "  sine "  methods  require  carefully  calibrated  instruments. 

§  15.  The  three  methods  are  seen  to  be  based  on  the  relation, 

power  volt-amperes 

total  volt-amperes 

wattless  volt-amperes 
sin  0  = 


total  volt-amperes 

wattless  volt-amperes 

tan  9  =  -  . 

power  volt-amperes 

§  1 6.  General  Method  for  Measuring  Power;  n — i  Watt- 
meters.— This  method  consists  in  selecting  any  one  conductor  of 
a  system  and  considering  it  as  a  common  return  for  all  the 
others.  One  wattmeter  is  then  used  for  each  conductor,  except 

*(§i4a).  With  non-sine  waves,  the  value  of  power  factor  by  the 
tangent  method  would,  theoretically,  be  a  little  larger  than  the  true  value 
by  the  cosine  method;  the  value  by  the  sine  method  would  be  a  little 
larger  than  the  value  by  the  tangent  method. 


6-B]  MEASUREMENT    OF    POWER.  227 

this  common  return.  No  wattmeter  is  required  for  a  return 
circuit ;  thus,  for  a  2-wire  system,  one  wattmeter  only  is  needed, 
no  wattmeter  being  needed  in  the  return  conductor;  in  a  3~wire 
system,  two  wattmeters  are  used,  none  being  needed  in  the  re- 
turn conductor,  etc.  If  n  is  the  number  of  line  conductors, 
n  —  i  wattmeters  are,  accordingly,  required.  For  a  3~wire  sys- 
tem, the  connections  are  shown  in  Fig.  i. 

To  measure  power  in  any  system,  connect  a  wattmeter  in  every 
line  circuit  except  one  (considered  as  the  return  conductor), 
each  wattmeter  having  its  current  coil  in  series  with  one  of  the 
lines  and  its  potential  coil  connected  from  this  line  to  the  return 
conductor.  One  less  wattmeter  is  required  than  the  number  of 
line  wires;  the  total  power  is  the  algebraic  sum  of  the  individual 
wattmeter  readings. 

§  17.  To  read  positive  power  each  wattmeter  is  to  be  connected 
in  the  positive  sense, — that  is,  connected  in  the  same  way  as  for 
measuring  power  in  a  2-wire  system,  direct  or  alternating.  If, 
when  connected  in  this  manner,  the  needle  of  any  wattmeter 
deflects  the  wrong  way,  the  connections  of  its  potential  or  current 
coil  are  to  be  reversed  and  its  reading  is  to  be  considered  negative. 
Compare  §  25. 

§  18.  This  method  of  measuring  power  is  absolutely  general; 
the  current  may  be  direct  or  alternating  and  may  vary  by  any 
law  whatsoever ;  the  system  may  be  single-phase  or  polyphase, 
balanced  or  unbalanced,  symmetrical  or  unsymmetrical. 

As  a  particular  case,  the  two-wattmeter  method  for  a  3-wire 
system  is  of  special  importance  with  reference  to  3-phase  circuits 
and  will  be  considered  later  (§23)  in  detail. 

§  19.  The  foregoing  method  -has  been  explained  by  considering 
one  cpnductor  as  a  common  return  for  all  the  others,  and  for 
most  purposes  this  explanation  is  sufficient.  The  method  with 
n — i  wattmeters  can  be  rigorously  established  (§22)  by  first 
developing  the  method  with  n  wattmeters,  §  20. 


228  POLYPHASE   CURRENTS.  [Exp. 

§20.  General  Method,  n  Wattmeters. — In  any  star-connected 
system,  if  a  wattmeter  is  connected  in  each  line — the  current 
coil  connected  in  series  with  the  line  and  the  two  ends  of  the 
potential  coil  connected,  respectively,  to  the  line  wire  and  to  the 
junction  or  neutral  point  of  the  system — the  total  power  of  the 
system  will  be  the  sum  of  the  separate  wattmeter  readings,  as 
discussed  in  §  2. 

§21.  This  arrangement  of  wattmeters,  however,  is  not  limited 
to  star-connected  circuits ;  nor  is  it  necessary  to  have  the  neutral 
point  accessible.  The  true  power  of  any  system  whatsoever  may 
be  measured  by  connecting  one  wattmeter  in  each  line,  with  cur- 
rent coil  in  series  with  the  line  and  potential  coil  with  one  end 
connected  to  the  line  and  the  other  end  to  any  point  P  of  the 
system,  which  may  or  may  not  be  the  neutral.  To  this  potential 
point  P  is  connected  the  potential  coil  of  every  wattmeter.  The 
algebraic  sum  of  the  wattmeter  readings  gives  the  true  power. 
A  general  proof  of  this  is  given  in  §  53 ;  it  can  be  verified  by  ex- 
periment, §§  45-49- 

§  22.  The  fact  that  any  point  of  the  system  may  be  taken  as 
the  potential  point  P  leads  to  the  practical  simplification  by  which 
one  wattmeter  is  omitted.  In  a  system  of  line  wires,  a}  b,  c  •  -  •  n} 
let  the  line  wire  n  be  taken  as  the  potential  point.  Wattmeters 
A,  B,  C,  etc.,  will  have  current  coils  connected  in  series  with 
a,  b,  c,  etc.,  and  potential  coils  connected  from  a  to  n,  from  b  to  n, 
etc.  Wattmeter  N  would,  accordingly,  have  its  potential  coil 
connected  from  n  to  «;  as  both  ends  of  the  pressure  coil  would 
thus  be  connected  to  the  same  point,  this  wattmeter  would  always 
read  zero  and,  accordingly,  can  be  omitted.  The  method  of 
n — i  wattmeters,  §  16,  is  thus  established. 

§  23.  Two  Wattmeter  Method  for  any  3-wire  system. — This 
is  the  method  generally  used  for  measuring  3-phase  power.  Be- 
ing a  particular  application  of  the  n — I  wattmeter  method, 
§  1 6,  the  two-wattmeter  method  can  be  applied  to  any  3-wire 


6-B]  MEASUREMENT    OF    POWER.  229 

system*  and  is  independent  of  any  assumptions  as  to  wave  form 
or  the  nature  of  the  load. 

§24.  The  arrangement  of  instruments  is  shown  in  Fig.  i.  The 
wattmetersf  are  inserted  in  any  two  lines,  as  X  and  Y ,  the  third 
wire  Z  being  considered  as  a  common  return. 


LJL 


» 

—  < 

—  « 

wl 

V 

[ 

j 

Y 

Wo 

1! 

FIG.    i.     Two-wattmeter   method    for  measuring    power    in    any    3-phase    or 
other  3-wire  circuit. 

The  total  power  is  the  algebraic  sum  of  the  readings  of  the 
two  wattmeters.  For  high  power  factors  (more  than  0.5)  this 
will  be  the  arithmetical  sum,  both  wattmeter  readings  being  posi- 
tive. For  low  power  factors  (less  than  0.5),  the  reading  of  one 
wattmeter  is  to  be  considered  negative,  the  total  power  in  this 
case  being  the  arithmetical  difference  of  the  two  readings,  as 
shown  later  in  §31. 

§  25.  There  are  several  ways  for  telling  whether  one  reading  is 
negative  or  not,  the  principal  ones  being  as  follows : 

(a)   From  the  sense  of  the  connections,  §  17. 

*  (§23a).  If  each  end  of  a  3-phase  line  has  its  neutral  well  grounded,  it 
becomes  virtually  a  4-wire  system;  the  ground  circuit  can  not  be  neglected 
unless  the  load'  is  practically  balanced. 

t  (§243).  Polyphase  Wattmeter. — Instead  of  two  single-phase  watt- 
meters, a  single  instrument  combining  the  two  is  commonly  used.  This 
consists  of  two  wattmeters,  one  above  the  other,  with  the  moving  elements 
mounted  upon  a  common  shaft.  The  reading  of  such  an  instrument  gives 
the  total  power.  The  electrical  connections  are  the  same  as  for  two 
separate  instruments. 


230  POLYPHASE    CURRENTS.  [Exp. 

(b)  For  the  given  load  substitute  a  load  that  is  non-inductive 
or  is  known  to  have  high  power  factor;  if,  with  certain  connec- 
tions, both  wattmeters  deflect  properly,  their  readings  for  these 
connections  are  positive.     When  one  connection  needs  to  be  re- 
versed to  obtain  proper  deflection,  one  reading  is  negative. 

(c)  Disconnect  one*  potential  circuit  from  the  middle  wire  Z 
and  connect  it  to  the  outside  wire,  X  or  Y ;  if  the  wattmeter  re- 
verses, the  readings  of  one  of  the  wattmeters  must  be  considered 
negative.  > 

Method  (c)  can  be  readily  applied  during  test,  when  using  the 
two-wattmeter  method  on  a  3-wire  system,  but  does  not  apply  to 
a  system  with  more  than  three  wires. 

Method  (a)  is  general  and  can  be  applied  to  a  system  with 
any  number  of  wires.  The  polarity  of  the  wattmeter  circuits 
may  be  marked  once  for  all,  instruments  of  one  make  being 
similar.  The  instruments  can  be  properly  connected  in  the  posi- 
tive sensef  in  advance  and  confusion  during  the  test  avoided. 

§26.  Two-wattmeter  Method  with  Balanced  Three-Phase 
Load. — As  has  been  already  stated,  the  two-wattmeter  method  is 
general  for  any  kind  of  3-wire  circuit.  Detailed  proof  for  each 
particular  case  is,  accordingly,  unnecessary.  A  discussion  of  its 
application  to  measuring  a  balanced  3-phase  load  will,  however, 
prove  instructive  as  an  illustration  and  will  serve  to  make  clear 
the  negative  reading  of  one  wattmeter  at  low  power  factors. 
Furthermore,  it  will  show  a  method  for  obtaining  3-phase  power 
factor. 

§  27.  Fig.  2  is  the  diagram  for  a  balanced  3-phase  load,  it  being 
assumed  that  currents  and  voltages  follow  a  sine  law.  For  unity 

*(§25a).  On  a  3-phase  circuit  it  is  sufficient  to  do -this  with  one 
potential  circuit  only;  but  in  general  it  should  be  done,  successively,  with 
each  potential  circuit,  a  reversal  of  either  instrument  indicating  that  one 
reading  is  negative. 

t(§2Sb).  This  also  indicates  the  direction  of  the.  flow  of  power;  see 
"  Polyphase  Power  Measurements,"  by  C.  A.  Adams,  Elect.  World,  p.  143, 
January  19,  1907. 


6-B] 


MEASUREMENT   OF    POWER. 


231 


power  factor  (0  =  o),  the  three  line  currents  are  shown  by  the 
heavy  arrows  /x,  /Y,  Iz.  The  dotted  arrows  show  these  currents 
for  lower  power  factors,  #  =  30°,  0  =  6o°  and  0  =  90°. 


FIG.  2.     Currents  and  voltages  in  a  balanced  3-phase   system. 

If  two  wattmeters  are  connected  as  in  Fig.  i,  wattmeter  (i) 
has  a  current  !x  in  its  current  coil  and  a  voltage  Exz  across  its 
potential  coil,  the  phase  difference  between  this  current  and 
voltage  being  6  —  30°.  The  component  of  Ix  in  phase  with  Exz 
is,  accordingly,  Ix  cos  (6  —  30°)  ;  hence — writing  E  for  Exz  and 
/  for  Ix — wattmeter  (i)  reads 

Wt  =  EI  cos  (0  —  3o°)=£7  (cos  30°  cos  0  + sin  30°  sin  0). 

In  a  like  manner,  wattmeter  (2)  has  a  voltage  EYZ  and  a  cur- 
rent /Y,  having  a  component  /Y  cos  (0  +  30°)  in  phase  with  EYZ. 
Hence — writing  E  for  Eyz  and  /  for  /Y — wattmeter  (2)  reads 

PF2  =  £/cos  (0  +  30°)=£7  (cos  30°  cos  0  —  sin  30°  sin  0). 
Adding  Wz  to  Wlf  we  have 

Wi+W2  =  2EI  cos  30°  cos  0  =  V3£/  cos  0, 


23  2 


POLYPHASE   CURRENTS. 


[Exp. 


which  is  seen  to  be  the  expression  for  the  total  power  in  a  3-phase 
circuit  (§§20,  21,  Exp.  6-A).  The  two-wattmeter  method  for 
a  balanced  3-phase  load  is  thus  established. 

§  28.  Power  Factor.  —  Subtracting  W2  from  Wlf  we  have 


W1—W2  =  2EI  sin  30°  sin  0  =  EI  sin  0. 
Hence,  by  dividing,  we  have    . 

Wi  —  EF2tan0 
~ 


The  value  of  0  and  of  power  factor  (cos  6)   for  a  balanced 
3-phase  circuit  is,  accordingly,  determined  by  the  tangent  formula 


The  larger  reading  is  Wlf  and  is  always  positive;  the  smaller 

reading,  Wz,  may  be  positive  or  negative. 

§  29.  To  save  labor  in  com- 
putation, it  is  convenient  to 
plot  a  curve,  Fig.  3,  with 
power  factor  (cos  6)  as  or- 
dinates  and  the  ratio  of  watt- 
meter readings,  Wz  -f-  Wlt  as 
abscissae.  Points  on  this  curve 
are  determined  by  the  relation 


jgative 


30 


By  means  of  this  curve,  the 


-1.00  -.SO    -.liO     -.40     -.20        0      +.iJO    +.40   ^ .60  +.80+J.OO 
Ratio  of  Wattmeter  Readings,  IF-f^  Wl 

FIG.    3.       Power    factor    of   balanced       POwer    fact°r    f°r    a    balanced 

3-phase  circuit  for  different  ratios  of      load     is     readily     determined 

wattmeter    readings    in    two-wattmeter 
method. 


from   the    ratio    of    the    two 
wattmeter  readings.   For  plot- 
ting the  curve  in  Fig.  3,  the  following  points  were  determined: 


6-B]  MEASUREMENT   OF    POWER.  233 

W**-Wi  —I.  —.80    —.60    —40    —.20       O   +.20     +40     +.60     +.80     +1. 

cos  0  o  .064  .143  .240  .359  .5  .655  .803  .918  .982  I. 

Intermediate  values  can  be  found  by  interpolation.  It  is  seen 
that  the  curve  is  not  symmetrical. 

§  30.  Errors  of  calibration  are  avoided  if  one  wattmeter  is 
used,  successively,  in  the  two  positions  to  determine  W±  and  W2. 
Since  the  assumption  is  made  that  the  current  in  the  wattmeter 
is  the  same  for  the  two  readings  (/X  =  /Y  =  /),  greater  accuracy 
is  obtained  if  the  current  in  the  two  cases  is  actually  the  same 
current.  This  is  accomplished  by  using  the  one  wattmeter  method 
of  Fig.  7,  which  is  more  accurate  for  determining  power  factor 
than  is  the  two-wattmeter  method.  In  either  case,  corrections 
may  be  made  (§42)  for  slight  variations  in  voltage. 

§31.  Negative  Reading  of  Wattmeter. — Referring  to  Fig.  2, 
it  is  seen  that  for  all  values  of  6  from  o  to  90°,  the  projections  of 
/Y  upon  Ezx  have  the  same  sign;  the  wattmeter  reading  W^  is, 
therefore,  in  all  cases  positive. 

The  projection  of  Iy  upon  EYZ  decreases  as  6  increases,  be- 
comes zero  when  0  =  6o°,  and  then  changes  sign.  The  watt- 
meter reading  W2t  accordingly,  changes  sign,  being  positive  when 
6  is  less  than  60°  (power  factor  more  than  0.5)  and  negative 
when  6  is  more  than  60°  (power  factor  less  than  0.5).  In  all 
cases  W±  is  the  larger,  and  W2  is  the  smaller,  reading. 

On  non-inductive  load,  6  =  0  and  IV^  =  W2',  each  wattmeter 
reads  half  the  total  power.  When  #  =  90°,  W^  =  —W2  and  the 
total  power  is  zero. 

§  32.  Three-Phase  Power  with  One  Wattmeter. — With  only 
one  wattmeter,  3-phase  power  can  be  measured  by  the  two-watt- 
meter method  (§23)  by  using  suitable  switches  for  throwing  the 
wattmeter  from  one  position  to  the  other.  This  procedure  gives 
the  true  power  for  unbalanced  as  well  as  balanced  loads  and  is 
generally  the  best  one  to  follow. 

The  transfer  of  the  current  coil  of  the  wattmeter  from  one 
line  to  another  is  not  always  convenient  or  possible  and,  when 


234  POLYPHASE   CURRENTS.  [Exp. 

•the  load  is  balanced,  the  power  in  a  3-phase  system  can  be 
measured  with  only  one  wattmeter  without  such  transfer  by  one 
of  the  following  methods. 

§  33-  With  Neutral  Available. — When  the  neutral  is  available, 
the  current  coil  of  the  wattmeter  can  be  connected  in  any  one 
line  circuit  and  the  potential  coil  connected  from  that  line  to  the 
neutral.  For  a  balanced  load,  the  total  power  will  be  three  times 
the  reading*  of  the  wattmeter.  The  power  factor  is  equal  to 
W —.-El,  where  /  is  the  line  current  and  E  is  the  star  voltage. 
When  the  load  is  not  balanced  the  total  power  will  be  the  sum  of 
three  readings,  one  on  each  phase. 

§  34.  With  Artificial  Neutral. — When  the  neutral  is  not  avail- 
able, an  artificial  neutral  can  be  created,  as  by  means  of  three 
equal  star-connected  non-inductive  resistances,  R^  R2,  R3  in 
Fig.  4.  The  method  of  §  33  can  then  be  applied. 

It  is  necessary  that  these  resistances  be  relatively  low,  as  com- 
pared with  the  resistance  of  the  potential  circuit  Rw  of  the 

wattmeter.  The  current  in 
them  will  then  be  relatively 
large,  so  that  the  potential  of 
the  neutral  will  not  be  dis- 
turbed by  the  connection  of 
the  potential  circuit  of  the 
FIG.  4.  Measuring  power  with  one  wattmeter.  The  power  taken 

wattmeter  connected  to  the  neutral  in  ;n  the  resistances  may  be  in- 
a  balanced  3-phase  circuit. 

eluded  or  not  in  the  measured 

power  as  desired ;  correction  for  this  power  can  be  made  when 
necessary. 

§  35.  Strictly  speaking  R{  and  R2  should  each  be  equal  to  the 
joint  resistance  of  R3  and  Rw  in  parallel.  In  this  case  there  is 
no  need  of  making  the  resistances  low ;  this  leads  to  the  method 
of  §  36  in  which  Rs  is  omitted  entirely,  that  is,  R3=  oo. 

*(§33a).  Calibration  for  Total  Power. — -In  this  method,  or  in  any 
method  depending  upon  a  single  reading,  the  wattmeter  can  be  calibrated 
to  read  total  power. 


6-B] 


MEASUREMENT   OF   POWER. 


235 


§  36.  With  Y  -Multiplier.  —  With  a  balanced  load  and  with  one 
wattmeter,  the  current  coil  of  the  wattmeter  is  connected  in  one 
line.  One  end  of  the  potential  circuit  is  connected  to  the  same 
line,  the  other  end  being  con- 
nected to  the  junction  of  two 
resistances  R±  and  R2,  which 
are  connected  to  the  other  two 
lines,  as  shown  in  Fig.  5.  The 
resistances  R:  and  R2  are  non- 
inductive  and  are  each  equal* 
to  Rw,  the  resistance  of  the 


X 

X 

Y        i 

t                 ^    Llij 

Y 

FIG.  5.  Measuring  power  with  one 
wattmeter  and  a  Y-multiplier  in  a  bal- 
anced 3-phase  system. 


potential  circuit  of  the  watt- 
meter. 

True  power  is  three  times  the  reading  of  the  wattmeter,  cali- 
brated as  a  single-phase  instrument.  The  resistances  are  some- 
times put  up  in  a  special  volt-box  or  Y-multiplier  for  3-phase  cir- 

cuits ;  the  instrument  may  then  be 
calibrated  so  as  to  read  total  power, 
§33a. 

§  37.  By  Means  of  T-connection.  — 
In  a  3-phase  system,  with  three  lines 
X,  Y,  Z,  connect  the  current  coil  of 
the  wattmeter  in  any  one  line,  as  Z, 
Fig.  6.  Connect  the  potential  coil 
from  Z  to  a  point  O,  the  middle 
point  of  a  transformer  coil  across  XY. 
See  Fig.  15,  Exp.  6-A.  In  any  balanced  3-phase  system,  how- 
ever the  load  is  connected,  the  wattmeter  will  now  give  one  half 
the  total  power.  (This  may  be  seen  as  follows:  If  the  wattmeter 

*(§36a).  Provided  Ri  and  R2  are  approximately  equal  to  each  other, 
this  same  method  may  be  used  without  having  R1  and  R2  equal  to  /?\v. 
The  instrument  is  calibrated  as  a  single-phase  wattmeter  with  R!  and  R2  in 
parallel  with  each  other  and  in  series  with  Rw,  a  single  reading  then 
gives  one  half  the  total  power.  Compare  §§  Qa,  33a. 


FIG.  6.  Measuring  power 
with  one  wattmeter,  !T-con- 
nected,  in  a  balanced  3- 
phase  circuit. 


I 
236  POLYPHASE   CURRENTS.  [Exp. 

were  connected  with  its  potential  coil  on  the  star  voltage,  the  watt- 
meter would  read  one  third  the  total  power;  with  its  potential 
increased  50  per  cent.  —  see  Fig.  2,  —  it  will  read  one  half  the  total 
power.) 

§  38.  The  power  factor  is  W  -i-  £oz/z.  The  power  factor 
can  be  found  from  the  tangent  formula,  by  taking  one  reading, 
WK  of  the  wattmeter  with  the  connections  as  described  and  a 
second  reading,  W2,  with  the  potential  circuit  of  the  wattmeter 
transferred  to  XY. 

W2        £XY/Z  sin  6     . 

n  ;  nence  tan  0  = 


§  39.  Two-reading  Method.  —  This  is  one  of  the  simplest  and 
most  satisfactory  methods  for  measuring  power  and  power  fac- 
tor with  one  wattmeter  in  a  balanced  3-phase  circuit.  The 
,  current  coil  is  connected  in  one  line,  as 

V 

Z,  Fig.  7,  one  end  of  the  potential  circuit 
being  connected  to  the  same  line.  The 
other  end  of  the  potential  coil  .is  con- 
nected, successively,  to  X  and  Y  '  ,  and  a 
Y  reading  taken  in  each  position.  The 


FIG.  7.  Measuring  power  algebraic  sum  of  the  two  readings  gives 
by  two  readings  of  one  the  total  power.  (The  smaller  readings, 
wattmeter  in  a  balanced  w  ^  considered  negative  whenever  it 

3-phase  circuit. 

is  necessary  to  reverse  the  potential  or 
current  coil  of  the  wattmeter  to  obtain  a  proper  deflection.) 

§  40.  The  proof  of  the  method  will  be  seen  by  referring  to 
Fig.  2,  which  assumes  that  voltages  and  currents  follow  a  sine 
law.  The  two  readings  of  the  wattmeter  are 

Wi  =  El  cos  (0  —  30° )  ;     W*  =  EIcos(e  +  30° ) . 

Hence,  the  sum  of  the  two  readings  gives  the  total  power,  §  27. 

§41.  The  power  factor  (cos0)  is  determined  from  the  tangent 
formula,  §  28, 


6-B]  MEASUREMENT   OF    POWER.  237 


By  referring  to  Fig.  3,  power  factor  can  be  found  directly 
from  the  ratio  W2  -i-  Wt. 

§  42.  When  there  is  an  appreciable  difference  between  the 
phase  voltages  (which  we  may  term  E^  and  E2)  across  which  the 
potential  circuit  is  connected  when  W±  and  Wz  are  read,  a  more 
accurate  value  of  power  factor  will  be  obtained  by  correcting 
W^  or  W2  by  direct  proportion  to  obtain  values  correspond- 
ing to  equal  voltages.  The  ratio  Wz  -=-  W^  then  becomes 
EJW2  —  EJV^  The  power  factor  thus  determined  is  quite 
accurate,  being  independent  of  the  calibration  of  any  instrument 
and  of  any  slight  inequality  in  the  phases.  Even  for  an  un- 
balanced load,  it  gives  accurately  the  value  of  cos  6  for  /z,  where 
0  is  the  phase  difference  between  /z  and  the  voltage  OZ  (Fig.  2) 
midway  in  phase  between  XZ  and  FZ.  The  method  is  more 
accurate  with  one  than  with  two  wattmeters,  §  28. 

§  43.  Power  Factor  by  Sine  Method.  —  The  power  factor  of  a 
balanced  3-phase  circuit  'can  be  determined  by  the  sine  method 
(§13)  with  only  a  single  reading  of  voltmeter,  ammeter  and 
wattmeter.  The  method  does  not  require  the  neutral  to  be 
available,  nor  does  it  require  any  auxiliary  resistances  or  other 
devices. 

Representing  the  three  line  wires  as  X,  Y  and  Z,  the  ammeter 
and  the  current  coil  of  the  wattmeter  are  connected  in  one  line, 
as  Z.  The  voltmeter  and  the  potential  coil  of  the  wattmeter 
are  connected  across  the  other  two  lines,  X  and  Y.  The  watt- 
meter reading  gives  the  wattless  or  quadrature  volt-amperes, 


sin  0, 
from  which  0  and  cos  0  are  determined. 


238  POLYPHASE   CURRENTS.  [Exp. 

PART   II.      MEASUREMENTS. 

§  44.  Many  of  the  methods  just  described  for  measuring  poly- 
phase power  and  power  factor  can  best  be  taken  up  as  occasion 
arises  for  their  use.  Without  undertaking  in  the  present  experi- 
ment to  subject  all  of  these  methods  to  test,  it  will  be  well  to 
select  a  few  of  them  for  trial  in  the  laboratory  in  order  to  illus- 
trate and  make  clear  the  methods  as  a  whole.  For  this  the 
following  tests  are  suggested. 

§  45.  Verification  of  Methods  for  Measuring  Polyphase  Power. 
— With  a  single-phase  non-inductive  load,  forming  a  2-wire  sys- 
tem, measure  the  total  power  with  two  wattmeters.  Each  line 
is  to  contain  the  current  coil  of  one  wattmeter,  the  potential  coil 
of  which  is  connected  from  the  line  to  a  common  point  P,  as  in 
§  21.  The  experiment  consists  in  connecting  P  to  different 
parts  of  the  circuit,  of  various  potentials,  and  noting  that  the 
algebraic  sum  of  the  two  wattmeter  readings  is  constant. 

When  the  power  indicated  by  one  wattmeter  becomes  greater, 
as  P  is  changed,  the  power  indicated  by  the  other  wattmeter  be- 
comes less.* 

§46.  For  example,  let  the  supply  lines  be  a^a^  as  in  Fig.  6, 
Exp.  6- A.  Connect  P,  successively,  to  points  of  different  po- 
tential, as  alt  a2,  the  neutral  O,  A^  and  A2,  these  points  being 
all  on  phase  A.  When  phase  B  of  a  two  phase  supply  is  avail- 
able, proceed,  also,  to  connect  P  successively  to  points  Blt  blf 
b2,  B2  on  phase  B. 

§  47.  Repeat  with  an  inductive  load. 

§  48.  Repeat  in  some  modified  manner,  as  by  using  ajb^  as 
supply  lines  and  connecting  P,  successively,  to  various  points  as 
described  above. 

§  49.  When  points,  as  in  Fig.  6,  Exp.  6- A,  are  not  available, 
a  resistance  can  be  bridged  across  the  circuit  and  the  point  P 

*  A  positive  reading  decreases ;  a  negative  reading  increases. 


6-B]  MEASUREMENT   OF   POWER.  239 

connected  to  different  points  on  this  resistance.  The  load  re- 
sistance itself  can  be  thus  utilized. 

The  experiment  might  be  extended  to  using  3  wattmeters  on 
a  3-wire  system,  4  wattmeters  on  a  4-wire  system,  etc.,  but  this 
seems  hardly  necessary.  The  method  of  n  wattmeters,  n  —  I 
wattmeters  and  two  wattmeters  may,  in  this  way,  be  experi- 
mentally verified. 

§  50.  Two-phase  Power  Factor. — From  one  phase,  A,  of  a 
2-phase  supply  draw  a  single-phase  load.  Take  measurements 
with  a  voltmeter,  ammeter  and  wattmeter  and  determine  the 
power  factor  by  the  "  cosine  method,"  §  14. 

Transfer  the  voltmeter  and  potential  coil  of  the  wattmeter 
to  the  other  phase,  B,  and  determine  the  power  factor  by  the 
"sine  method,"  §  13,  and  by  the  "tangent  method,"  §  12. 

§51.  Three-phase  Power  and  Power  Factor. — With  a  3-phase 
balanced  load  supplied  from  a  3-phase  circuit,  take  two  readings 
of  a  wattmeter  connected  as  in  Fig.  7.  Determine  the  total 
power ;  calculate  the  power  factor  by  the  tangent  formula,  §  28, 
and  by  the  ratio  of  wattmeter  reading,  Fig.  3. 

§  52.  Transfer  the  potential  coil  of  the  wattmeter  to  the  third 
phase,  so  as  to  read  the  "  quadrature "  volt-amperes ;  take  the 
necessary  readings  of  the  wattmeter,  voltmeter  and  ammeter,  and 
determine  power  factor  by  the  "  sine  method,"  §  43. 


APPENDIX   I. 

MISCELLANEOUS   NOTES. 

§  53.  General  Proof. — In  any  system,  with  any  number  of  con- 
ductors a,  b,  c,  etc.,  let  the  instantaneous  values  of  the  currents  in 
these  conductors  be  ia,  t'B,  ic,  etc.  Designate  by  ea,  eb,  ec,  etc.,  the 
instantaneous  values  of  the  potentials  of  the  several  conductors.  The 
currents  and  electromotive  forces  may  vary  in  any  manner  what- 
soever. There  is  no  limitation  as  to  the  arrangement  or  method  of 
connection  of  the  generator  and  receiver  circuits. 


240  POLYPHASE   CURRENTS.  [Exp. 

» 

The  total  power  at  any  instant  is 

(1)  w  —  eaia  +  ebib  +  ecic  .  .  .  =2ei. 

Let  ep  be  the  instantaneous  potential  of  any  point  P  of  the  system. 
Since  it  is  known  that  ^i  =  o,  it  follows  that 

(2)  epia  +  epii>  +  epic  .  .  .  =  *P2«  =  o. 

Since   (2)   is  equal  to  zero,  it  may  be  subtracted  from   (i)   without 
affecting  its  value;  hence 

(3)  w=  (*o—  -*p)*'o  +  Ob  —  *p)*&  +  (ec  —  ev}ic  .  .  =  S(<?  —  ep)i. 
The  total  power  at  any  instant  is  seen  to  be  the  sum  of  the  products 
of  the  instantaneous  currents  in  each  conductor  and  the  instantaneous 
differences  of  potential   between  the  respective   conductors   and  the 
point  P. 

The   mean   power    W  is   found   by   integrating   the   instantaneous 
power  over  a  time  equal  to  one  period,  T,  and  dividing  by  T. 


f  JT  «v- 


But  each  one  of  these  terms  represents  the  power,  as  read  by  a 
wattmeter  with  current  coil  in  series  with  one  conductor  and  with 
potential  coil  connected  from  that  conductor  to  the  common  point  P, 
and  the  total  power  is  the  sum  of  the  several  wattmeters  so  con- 
nected. For  an  w-wire  system,  n  wattmeters  are  required,  the  total 
power  being 


When  the  point  P  coincides  with  one  conductor,  the  wattmeter  for 
that  conductor  reads  zero  and  can  be  omitted  ;  n  —  i  wattmeters  are 
then  required. 

The  method  for  n  wattmeters,  for  n  —  i  wattmeters,  and  for  two 
wattmeters,  is,  accordingly,  proved  without  reference  to  wave  form 
or  the  character  of  the  load.  This  general  proof  was  first  given 
by  A.  Blondel,  p.  112,  Proceedings  International  Electrical  Congress, 
Chicago,  1893. 


CHAPTER   VII. 

PHASE    CHANGERS,    POTENTIAL    REGULATORS,    ETC. 
EXPERIMENT  7-A.   Polyphase  Transformation. 

§  i.  Possible  Kinds  of  Transformation. — The  transmission  of 
power  in  a  single-phase  system  is  pulsating,  no  matter  what  the 
character  of  the  load.  This  can  be  readily  seen  by  sketching 
assumed  curves  for  the  instantaneous  values  of  an  alternating 
electromotive  force  and  current,  and  plotting  the  products  of  the 
ordinates  from  instant  to  instant  as  a  power  curve.*  In  a  bal- 
anced polyphase  system,  however,  the  pulsations  of  power  in  the 
different  phases  are  seen  to  so  combine  that  the  total  transmis- 
sion of  power  is  uniform,f  without  pulsation.  (See  §2,  Exp. 
6-A.) 

§  2.  Polyphase  to  Single-phase  Transformation  not  Possible. — 
In  a  transformer,  neglecting  the  slight  modification  due  to  losses, 
the  power  given  into  the  primary  at  any  instant  is  equal  to  the 
power  taken  out  of  the  secondary  at  that  instant.  It  is  not  pos- 
sible, therefore,  simply  by  means  of  transformers  to  change  a 

*  (§  ia).  The  area  included  between  the  power  curve  and  time  axis  rep- 
resents energy,  this  energy  being  positive  (supplied  to  the  line)  or  nega- 
tive (returned  from  the  line)  according  to  whether  the  current  and  electro- 
motive force  have,  at  the  time,  like  or  unlike  signs,.  It  is  instructive  to 
sketch  curves  for  currents  differing  in  phase  from  the  electromotive  force 
by  o,  45  and  90  degrees. 

f  (§  ib).  This  can  be  shown  for  a  2-phase  system  by  drawing,  for  each 
phase,  sine  curves  for  electromotive  force  and  current  and  plotting  the 
product  as  a  power  curve.  Two  power  curves  are  thus  obtained,  one  for 
each  phase,  and  it  will  be  seen  that  the  crests  of  one  correspond  to  the 
hollows  in  the  other,  the  algebraic  sum  of  the  two  power  curves  being 
constant.  The  sum  of  the  three  power  curves  for  a  3-phase  circuit  can 
be  shown  to  be  constant  in  the  same  way. 
17  241 


242  PHASE   CHANGERS,   ETC.  [Exp. 

pulsating  into  a  non-pulsating  transmission  of  power,  or  vice 
versa. 

It  is  accordingly  not  possible,  by  means  of  transformers,  to 
draw  single-phase  current  from  a  polyphase  system  and  draw 
from  the  several  phases  equally,  so  that  the  flow  of  energy  is 
non-pulsating.  To  accomplish  such  a  transformation,  use  is 
made  of  a  motor-generator*  consisting  of  a  polyphase  motor 
driving  a  single-phase  generator.  The  moving  parts  act  as  a 
flywheel,  storing  and  restoring  kinetic  energy,  thus  accounting 
for  the  momentary  differance  between  the  pulsating  output  and 
non-pulsating  input  of  electric  energy.  This  method  is  advo- 
cated for  running  single-phase  railway  feeders  from  a  polyphase 
transmission  line. 

§  3.  Single-phase  to  Polyphase  Transformation  not  Possible. — 
It  is  likewise  not  possible,  by  means  of  transformers,  to  change 
a  single-phase  into  a  balanced  polyphase  system.  This  too  can 
be  done  by  means  of  a  motor-generator,  or  by  running  a  poly- 
phase induction  motor  on  a  single-phase  circuit, — a  2-phase  or  a 
3-phase  motor  giving  2-phase  or  3-phase  currents.  Various  sta- 
tionary phase-splitting  devices  will  give  difference  in  phase  suffi- 
cient for  starting  induction  motors  on  single-phase  circuits,  but 
such  devices  cannot  give  balanced  polyphase  currents. 

§  4.  Polyphase  Transformation  Possible. — It  is  possible,  how- 
ever, by  various  arrangementsf  of  transformers  to  change  from 
one  balanced  polyphase  system  to  any  other  balanced  polyphase 
system,  the  flow  of  energy  in  each  system  being  uniform.  This 
is  termed  polyphase  transformation  and  its  study  is  the  object 
of  this  experiment.  The  various  methods  of  polyphase  trans- 
formation are  similar  in  principle,  use  being  made  of  the  factf 

*  Such  a  motor-generator  has  been  installed  in  the  chemical  laboratory 
of  Cornell  University  to  supply  2,000  or  3,000  amperes  of  single-phase 
current  for  the  electric  furnace. 

t(§4a).  Two  transformers,  only,  are  necessary;  but  more  than  two 
are  used  in  some  arrangements,  as  Fig.  4. 

$  Fully  discussed  in  Exp.  6-A. 


7-A] 


POLYPHASE   TRANSFORMATION. 


243 


that,  if  two  coils  with  electromotive  forces  differing  in  phase  are 
connected  in  series,  the  electromotive  force  across  the  two  coils 
is  the  vector  sum  (or  difference)  of  the  two  separate  electro- 
motive forces.  A  resultant  electromotive  force  of  any  desired 
phase  can  thus  be  obtained  from  a  polyphase  supply  by  means 
of  two  transformers. 

The  transformation  from  2-phase  to  3-phase,  or  vice  versa,  is 
most  important  on  account  of  the  copper  economy*  in  3-phase 
transmission  and  the  sometime  advantage  of  2-phase  generation 
or  utilization. 

§  5.  Two-phase  to  Three-phase  Transformation  (and  vice 
versa)  by  T-connection. — This  method,  first  published  by  Mr. 
C.»  F.  Scott,  is  shown  diagrammatically 
in  Fig.  i,  in  which  A  and  B  are  the  two 
phases  of  a  2-phase  system ;  X,  Y  and  Z 
represent  the  three  line  wires  of  a 
3-phase  system. 

Let  the  transformation  be  from  2- 
phase  to  3-phase.  Two  transformers  are 
used.  One  has  a  primary  AA'  on  phase 
A  of  the  2-phase  system  and  has  a  sec- 
ondary (XY)  wound,  let  us  say,  for  100 
volts  with  a  middle  tap  at  O,  dividing  the 
coil  into  two  parts  of  50  volts  each. 


TWO  PHASE 
PHASE  A 


CO  CO 
< 


OB- 


FIG.  i.  Transformation 
from  2-phase  (AB)  to  3- 
phase  (XYZ),  or  vice 

The  second  transformer  has  a  primary 

BB'  on  phase  B  of  the  2-phase  system  and  has  a  secondary 
(OZ)  wound  for  86.6  volts  (86.6=iooX  IVs),  which  has 
one  end  connected,  as  shown,  to  the  middle  tap  of  the  first  trans- 
former. It  will  be  found  that  the  three  voltages,  XY,  YZ  and 
ZX,  are  equal  and  differ  in  phase  from  each  other  by  120°,  thus 
making  a  3-phase  system.  These  voltages  are  represented  in 
Fig.  2.  They  should  be  interpreted  as  in  Exp.  6-A;  see  also 
Appendix  I.  to  this  experiment. 
*  See  Appendix  III.,  Exp.  6-A. 


244  PHASE   CHANGERS,   ETC.  [Exp. 

§  6.  This  method  of  transformation  is  reversible ;  i.  e.,  if  a 
3-phase  system  be  connected  (see  Fig.  i)  to  XYZ  as  primary, 
2-phase  circuits  may  be  taken  from  A  A'  and  BB'  as  secondary. 
§  7.  Double    Transformation. — In  Fig.   3   is   shown  a   double 
transformation,  from  the  2-phase  gener- 
ating  circuits  A,  B  to  the  3-phase  trans- 
mission   circuits    X,    Y,    Z,    and    from 
these  to  the  2-phase  receiving  circuits  A, 
B.     The   receiving   circuits,   A    and   B, 
may  be  used  together,  as  for  operating 
FIG.  2.    Voltage  and  cur-     poiyphase  motors,  or  separately  as   for 

rent  relations. 

lighting. 

§  8.  As  a  further  explanation  of  the  T-connection,  referring 
to  Fig.  3,  suppose  the  connections  OZ',  OZ'  were  left  out  and 
that,  instead,  a  fourth  wire  zr  (not  shown)  were  used  to  connect 
Z'  and  Z' ;  each  phase  would  then  have  its  independent  2-wire 
transmission  circuit, — wires  xy  for  phase  A  and  zz*  for  phase  B. 
In  making  the  T-connection  of  Fig.  3,  the  fourth  wire  z'  is 
omitted*  and  in  its  place  use  is  made  of  the  two  wires  x  and  y, 
acting  in  parallel  as  a  single  conductor.  The  current  from  the 
coil  ZZ'  flows  to  O  and  divides,  passing  through  OX  and  OY 
differentially,  so  as  to  have  no  magnetizing  effect  on  the  core 
of  XY.  With  respect  to  the  current  from  Z',  the  two  parts  of 
the  coil  XY  are  wound  non-inductively.  They  should  be  inter- 
spaced so  as  to  have  the  least  possible  magnetic  leakage  and  con- 
sequent leakage  reactance,  which  would  give  poor  regulation  on 
phase  B.  This  precaution  is  necessary  in  winding  any  T-con- 
nected  transformer. 

The  regulation  of  phase  A  and  of  phase  B  are  as  independent 
of  each  other  with  three  wires  (Fig.  3)  as  they  would  be  with 
four  wires  making  separate  circuits ;  phase  B  may  have  a  heavy 

*(§8a).  There  is  obvious  copper  economy  in  this  case  in  changing 
from  a  4-wire  2-phase  to  a  3-wire  3-phase  transmission ;  see  Appendix  III., 
Exp.  6-A. 


7-A]  POLYPHASE   TRANSFORMATION.  H5 

motor  load  with  50  per  cent,  drop,  while  A  has  a  lighting  load 
with,  say,  2  per  cent,  drop,  unaffected  by  the  starting  and  stop- 
ping of  the  motors  on  B.  They  are  absolutely  independent  of 
each  other. 

§  9.  Composite  Transmission. — If  the  phases  A  and  B,  Fig.  3, 
were  generated  and  utilized  separately,  it  would  not  be  necessary 

for  B  to  differ  from  A 
by  ninety  degrees;  B 
could  have  any  phase, 
even  the  same  phase  as 
t  ^ "  A.  Again  A  and  B 

FIG     3.      Two-phase    generator    and   receiving      might     be     of     different 
circuits  with  3-phase  transmission. 

frequencies ;  in  fact  they 

can  be  treated  as  two  independent  transmission  systems*  whether 
of  the  same  or  of  different  frequencies.  In  the  same  way  a  direct 
and  alternating  current  can  be  combined  with  economy  of  copper 
and  independence  of  regulation. 

§  10.  Test. — First  note  the  single-phase  transformations  which 
can  be  made  with  the  transformers  to  be  used,  and  determine 
whether  or  not  the  transformers  are  suitable  for  the  purpose. 
Connect  the  transformers  so  as  to  transform  from  a  2-phase 
system  to  a  3-phase  system  and  make  measurements  of  the 
primary  and  secondary  line  voltages,  and  the  voltage  of  the 
T-connected  coil,  checking  all  by  computation. 

Make  corresponding  transformation  from  a  3-phase  to  a 
2-phase  system. 

If  the  transformers  are  provided  with  two  sets  of  coils,  for 
parallel  and  series  connection,  make  the  polyphase  transforma- 
tions with  all  possible  voltage  ratios.  Compute  the  volt-ampere 

*(§9a).  Various  methods  of  composite  transmission  will  be  found  in 
the  following:  Elect.  World,  February  28,  1903,  pp.  347  and  351,  Vol.  XLI., 
No.  9;  Am.  Electrician,  April,  1903,  pp.  189  and  177,  Vol.  XV.,  No.  4; 
Elect.  Review  (New  York),  March,  1903,  p.  362,  Vol.  42,  No.  n;  Elect. 
Age,  March,  1903,  p.  179,  Vol.  XXX.,  No.  3;  Mill  Owners,  April,  1903, 
p.  14. 


246  PHASE   CHANGERS,   ETC.  [Exp. 

capacity  for  the  3-phase  side  of  each  transformer  when  the  total 
power  output,  on  non-inductive  load,  is  100  watts ;  see  §  24, 
Exp.  6-A. 

§  ii.  Instructions  for  Special  Transformers. — These  instructions 
relate  to  two  transformers,  DEF  and  a/8yS.  Each  transformer  has 
two  primaries,  which  may  be  connected  in  series  or  in  parallel.  The 
windings  are  as  follows : 

Primary  a,  no  (or  165)  volts.          Primary!},  no  (or  165)  volts. 
Primary/?,  no  (or  165)  volts.          Primary  E,  no  (or  165)  volts. 
Secondary  y,  36!  (or  55)  volts.        Secondary  F,  63.5  (or  95.25)  volts. 
Secondary  8,  363  (or  55)  volts. 

The  first  number  gives  normal  voltage  for  highest  efficiency  at  60 
cycles ;  the  number  in  parenthesis  is  50  per  cent,  above  normal  voltage. 
These  transformers  were  specially  made  for  use  at  either  voltage. 

With  D  and  E  in  parallel  on  one  phase,  and  a  and  /3  in  parallel  on 
the  other  phase  of  a  2-phase  system,  connect  F  to  the  middle  point  of 
y  and  8  connected  in  series,  thus  making  a  T-connection.  From 
2-phase  circuits  of  no  volts  (and.  also  165  volts)  obtain  3-phase 
secondary  voltages  by  computation  and  measurement.  Make  the 
primary  connections  from  iio-volt  4-wire  2-phase  system,  and  also 
from  no-volt  3-wire  2-phase  system. 

Repeat  with  primaries  in  series  instead  of  in  parallel ;  compute  and 
measure  secondary  3-phase  voltages. 

Perform  corresponding  transformation  from  iio-volt  3-phase  to 
2-phase.  What  two  2-phase  voltages  can  be  thus  obtained? 

§  12.  Monocyclic  Transformation. — In  the  monocyclic  system 
(no  longer  being  installed)  a  single-phase  voltage  is  combined 
with  a  quadrature  voltage  of  one  fourth  its  value ;  thus,  in  Fig.  6, 
Exp.  6-A,  a  monocyclic  voltage  is  obtained  from  A^A2b2.  It 
is  an  unsymmetrical  2-phase  system.  If  two  I :  I  transformers 
are  used,  the  primary  of  one  being  connected  to  AJ)2t  of  the 
other  to  A2b2,  the  secondaries  (with  two  ends  together  for  a 
3-wire  system)  will  give  a  monocyclic  voltage  the  same  as  the 
primary.  This  secondary  voltage  is  an  open  delta  with  one  side 
reversed.  Test  this  with  a  voltmeter  and  draw  a  diagram  of 
voltages. 


7-A] 


POLYPHASE   TRANSFORMATION. 


247 


Reverse  the  primary  or  the  secondary  coil  of  one  transformer; 
the  secondary  voltages  now  form  an  open  delta,  making  very 
nearly  an  equilateral  triangle,  and  hence  forming  a  nearly  sym- 
metrical 3-phase  system.  Compute  these  voltages  and  verify  by 
measurement.  This  method  was  used  for 
obtaining  polyphase  current  for  operating 
3-phase  motors  upon  what  was  initially  a 
single-phase  circuit  with  a  so-called  "  tea- 
zer"  circuit  (b2)  added.  It  was  introduced 


by  Steinmetz;  the   name   indicated   a  pul- 


(XOY) 

(AOB). 


to      2-phase 


A  B 

FIG.    4.      Transforma- 

sating  (monocyclic)  flow  of  energy  instead    tion    from    3-phase 
of  a  non-pulsating  (polycyclic)  flow.    The 
transformation  is  instructive,  even  though 
its  introduction  has  been  discontinued. 

§  13.  Miscellaneous  Transformations. — Several  other  trans- 
formations are  here  indicated.  Try  these  by  experiment,  so  far 
as  time  and  facilities  permit.  Compute  for  each  case  the  excess 
of  volt-amperes  over  watts ;  see  §  24,  Exp.  6-A. 

§  14.  Fig.  4  shows  a  method  for  transforming  from  3-  to  2- 
phase,  with  three  transformers.     The  primaries  are  connected  to 
a  3-phase  supply.    The  secondaries  of  the  first 
^X^  „  two  transformers  are  OX  and  OY.    The  third 

transformer  has  two  secondaries,  whose  vol- 
tages are  AX  and  YB.  What  must  be  the 
values  of  these,  in  order  that  AOB  will  give 
true  2-phase  voltages? 

§  15.  Fig.  5  shows  a  method,  occasionally 
of  laboratory  use,  of  using  three  auto-trans- 
formers for  3-phase  transformation  from 
XYZ  to  xyz.  What  ratio  of  transformation 
will  be  obtained  by  using  three  auto-transformers  with  a  middle 
tap? 

§  16.  Fig.  6  shows  two  F-connected  auto-transformers  for 
3-phase  transformation  from  XYZ  to  Xyz.  This  is  a  method 


z 

FIG.     5. 
formation 


Trans- 
from    3- 


phase   (XYZ)  to  3- 
phase   (xyz). 


248  PHASE   CHANGERS,   ETC.  [Exp. 

commonly  used  for  obtaining  a  low  starting  voltage  for  3-phase 
motors  and  converters.     The  taps  y  and  z  can  be  located  where 
desired.     It  is  to  be  noted  that  the  voltage  is  changed,  but  not 
the    phase.      A    third    auto-transformer,    FZ, 
might   be   used,   with   a  tap  at   O.     Although 
better    for    continuous    operation,    this    would 
have  the   disadvantage   of   requiring  an   addi- 
"o"       *z,      tional  auto-transformer;  furthermore,  this  ar- 
FIG.  6.     Auto-     rangement  could  not  give  less  than  half  vol- 

transformers  on  3-      tagCj    an(j    would    make    a    reversal    of    phase    in 
phase  circuit. 

changing  from  low  to  high  (starting  to  run- 
ning) voltage,  which  is  not  desirable  in  starting  a  synchronous 
machine  from  the  alternating  current  end. 

§  17.  Two-phase  to  Six-phase  Transformation.— The  most 
practical  method  for  this  consists  in  transforming  from  a  2-phase 
primary  circuit  to  two  sets  of  T-connected  secondaries,  one  set 
being  inverted;  (thus  T  and  X).  Two  or  four  transformers  can 
be  used.  It  is  not  necessary  to  make  this  transformation  in  the 
laboratory.  (Detailed  connections  are  given  in  McAllister's 
Alternating  Current  Motors;  see  also  §  27,  Exp.  6-A.) 


APPENDIX  I. 
MISCELLANEOUS    NOTES. 

§  1 8.  Further  Interpretation  of  T-connection.— A  general  discussion 
of  the  vector  combination  of  electromotive  forces  is  given  in  Exp. 
6-A  (particularly  Appendix  I.),  and  the  general  principles  there  given 
can  be  applied  to  the  T-connection.  The  following  is  a  more  detailed 
discussion  of  this  particular  case. 

The  electromotive  force  of  any  alternating  current  coil  may  be  repre- 
sented by  a  vector  in  a  certain  direction.  If  this  coil  is  the  secondary 
of  a  transformer  connected  to  a  secondary  line,  as  in  the  present  case, 
the  electromotive  force  impressed  upon  this  line  will  be  represented 
by  the  same  vector.  If  in  connecting  the  coil  to  the  line  the  termi- 
nals are  reversed,  the  vector  representing  the  electromotive  force 


7-A] 


POLYPHASE   TRANSFORMATION. 


249 


100 


YO 


FIG.  7.  Two  senses 
in  v/hich  vectors  can 
be  considered. 


XY 


impressed  upon  the  line  is  likewise  reversed.     Thus,  in  Fig.  i.  the 

electromotive  force  of  the  coil  XY  may  be  a  vector  XY ;  when  it  is 

connected  in  the  opposite  sense,  the  electro- 
motive force  of  the  coil  YX  is  the  vector  YX. 

Fig.  7  shows  these  vectors  for  the  secondary     OY 

coils  of  Fig.  I.     From  these  elementary  prin-     Yx 

ciples   can  be   shown   the   delta   and   the   star 

equivalents  of  a  T-connection. 

§  19.  Delta   Equivalent    of    T-connection. — 

The  electromotive  forces  between  the  terminals 

XYZ  of  Fig.  I  or  Fig.  2,  should  be  considered 

in   a   certain   order,  XYZ   or  ZYX.     Let   us   consider   them   in  the 

XYZ  order,  as  shown  in  Fig.  8.  Going  from  X  to  Y,  we  have  the 
vector  XY  as  shown.  From  Y  to  Z,  we  have 
the  vectors  YO  and  OZ  (compare  Fig.  7) 
which  combine  to  give  YZ.  From  Z  to  X, 
we  have  the  vectors  ZO  and  OX  which  com- 
bine to  give  ZX.  The  three  resultant  vectors 
are  thus  shown  to  be  equal  and  to  differ  in 
phase  by  120°.  The  line  voltage,  thus  obtained 
by  the  T-connection,  is  accordingly  the  same 
as  would  be  obtained  by  three  3-phase  genera- 
tor coils,  XY,  YZ,  ZX,  connected  in  delta. 
§  20.  Star  Equivalent  of  T-connection. — Suppose  the  neutral  point 

N   (either  actual  or  imaginary)    in  the  coil  OZ  divides  its  voltage 

into  J  and  I ;  thus,  in  Fig.  9,  we  have  ON  = 

28.9  and  NZ  =  57.7,  with  arrows  down ;  NO 

=  28.9,  with  arrow  up  as  drawn.     From  the 

neutral  N  we  have  the  vector  NY,  the  resultant 

of  NO  and  OF;  and  NX  the  resultant  of  NO 

and  OX.     It  follows  that  NX,  NY  and  NZ  are 

each   equal  to   57.7  volts    (57.7=  ioo-=-  V3) 

and    differ    in    phase    from    each    other    by 

120°.     The    line    voltage,    thus    obtained    by 

the    T-connection,   is   the    same   as   would   be 

obtained  by  three  3-phase  generator  coils  NX,  NY,  NZ,  when  star- 
connected. 


FIG.  8.     Delta  equivalent 
of   T-connection. 


FIG.  9.    Star  equivalent 
of  T-connection. 


250  PHASE   CHANGERS,   ETC.  [Exp. 

EXPERIMENT  7-B.    Induction  Regulators.* 

§  i.  Types  of  Potential  Regulators. — When  a  generator  sup- 
plies current — either  alternating  or  direct — to  a  single  line,  the 
desired  voltage  at  a  distant  receiver  can  be  maintained  by  vary- 
ing the  excitation  of  the  generator,  either  by  a  hand-operated 
rheostat  or  by  some  automatic  device  as  the  Tirrell  regulator 
(§3a,  Exp.  i-B).  This  is  also  accomplished,  to  a  certain  ex- 
tent, by  a  compound  winding  (§4,  Exp.  i-B)  or  composite 
winding  (§  ua,  Exp.  3~A)  on  the  generator. 

When,  however,  a  generator  (or  several  generators  in  paral- 
lel) supplies  several  lines  or  feeders,  with  independently  varying 
loads,  this  simple  method  of  regulation  is  no  longer  possible; 
for  at  any  particular  time  the  voltage  on  one  feeder  may  be  too 
high,  while  the  voltage  on  another  feeder  is  too  low,  and  there 
is  no  change  of  station  voltage  which  can  be  made  which 
will  bring  the  delivered  voltages  on  all  feeders  to  their  proper 
values. 

In  direct  current  distribution  systems,  the  proper  voltage  can 
be  approximately  maintained  by  inter-connecting  the  various 
feeders  and  proportioning  the  amounts  of  copper  according  to 
average  load  conditions.  In  large  stations,  a  step  further  is 
taken  by  maintaining  in  the  station  several  sets  of  bus  bars  at 
different  voltages,  so  that  feeders  may  be  supplied  with  the 
proper  voltage  according  to  conditions,  long  feeders  being  sup- 
plied with  a  higher  voltage  than  short  ones.  Use  is  also  made 
of  auxiliary  batteries,  motor-driven  boosters,  etc.  In  early  sta- 
tions, wasteful  series  resistances  in  each  feeder  were  sometimes 
used. 

§  2.  In  an  alternating  current  system,  the  most  satisfactory 
resultsf  are  obtained  by  the  use  of  a  potential  regulator  in  each 

*  Note  that  an  induction  motor  with  wound  secondary  can  be  used  as 
an  induction  regulator ;  see  §  4. 

t(§2a).  Series  resistances  and  reactances  have  been  used  for  this  pur- 
pose. To  use  the  former  is  not  good  practice  on  account  of  the  energy 


7-B]  INDUCTION   REGULATORS.  251 

feeder.  The  potential  regulator  is  a  variable-ratio  transformer 
or  auto-transformer  used  to  raise  the  voltage  as  a  booster,  or 
to  lower  the  voltage  as  a  negative  booster.  The  regulator  may 
be  operated  either  manually,  or  automatically  by  means  of  a 
small  motor  which  is  controlled*  by  potential  wires  from  any 
desired  point  in  the  system. 

Alternating  current  potential  regulators  are  of  two  types,  the 
step-by-step  regulator  and  the  induction  regulator. 

§  3.  The  Step-by-step  Regulator. — The  step-by-step  regulator 
is  an  auto-transformer  (or  transformer)  with  switching  arrange- 
ments for  changing  the  number  of  turns.  In  principle  it  is  the 
same  as  any  auto-transformer  (Exp.  5-A).  In  practice,  the 
switching  arrangement  may  consist  of  a  number  of  individual 
knife  switches,  but  more  usually  is  either  of  a  drum  or  a  dial 
pattern,  operated  either  manually  or  automatically.  In  the  drum 
or  dial  pattern,  resistance  or  reactance  in  the  contact  leads  is 
sometimes  used  so  that  the  contact  arm  can  temporarily  bridge 
two  contact  points  without  disastrous  short  circuit. 

The  step-by-step  regulator  is  not  easily  made  automatic.  The 
contacts  deteriorate,  even  when  arcing  tips  are  used,  and  hence 
this  type  of  regulator  is  better  for  occasional  than  for  constant 
adjustment.  For  continuous  automatic  adjustment,  the  induc- 
tion regulator  is  generally^  used. 

§  4.  The  Induction  Regulator. — An  induction  regulator  is  a 
stationary  transformer  with  a  movable  primary  or  secondary 
which  may  be  set  in  different  positions  for  obtaining  potentials 

wasted.  Reactances  are  satisfactory  for  some  cases ;  to  be  effective,  how- 
ever, they  must  be  large  and  expensive. 

*  (§2b).  The  motor  may  be  either  direct  or  alternating  and  is  usually 
controlled  through  a  relay,  one  form  ot  Tirrell  regulator  being  made  for 
this  purpose. 

1"(§3a)-  In  cases  where  very  rapid  continuous  automatic  adjustment 
is  required,  the  induction  type  can  not  be  used  on  account  of  the  heavy 
mass  to  be  moved.  An  automatic  regulator  of  the  step-by-step  type  is 
better  for  this  rapid  adjustment,  because  the  moving  part  is  only  a  light 
contact  arm. 


252  PHASE   CHANGERS, '  ETC.  [Exp. 

of  different  values  or  of  different  phase.     A  form*  of  apparatus 
in  common  use  is  essentially  an  induction  motor  with  wound 
secondary  brought  out  to  terminals,  and  any  induction  motor  so 
constructed  can  be  used  as  an  induction  regulator. 
Such  an  apparatus  may  be  used : 

1.  As    a    single-phase    potential    regulator;    used    on    lighting 
feeders. 

2.  As  a  phase  shifter;  used  in  laboratory  testing. 

3.  As  a  polyphase  potential  regulator ;  used  on  polyphase  lines, 
particularly  in  supplying  current  to  synchronous  converters. 

§5.  (i)  Single-phase  Potential  Regulator. 
— Supply  the  primary   (or  one  phase  of  the 
primary   if    a   polyphase   induction   motor   is 
used)  with  a  constant  single-phase  voltage  not 
exceeding  the  normal  voltage  of  the  apparatus. 
FosUuxn  of  Rator         The  secondary  voltage  may  be  varied  by  turn- 
FIG.   i.     Use   as  ing  the  rotor  by  hand  to  any  desired  position, 

single-phase     trans-    ^  current  be  drawn          tQ  the  full.load 

iormer ;     secondary 

voltage    for   differ-  rating  of   the   secondary.     The   apparatus   is 

ent  positions  of  ro-  useci  m  two  ways :  (a)  as  a  transformer  with 

primary  and  secondary  not  connected  together ; 

(&)   as  an  auto-transformer  with  the  two  coils  connected,  as  in 

Fig.  2. 

§  6.  (a)  Use  as  Transformer. — Place  a  voltmeter  across  the 
open  secondary  and  revolve  the  rotor  step  by  step,  so  that  the 
secondary  potential  changes  between  zero  and  a  maximum.  (On 
open  circuit  the  data  for  methods  (a)  and  (b)  can  be  taken 

*  (§4a).  An  earlier  form  of  regulator  had  stationary  primary  and  sec- 
ondary coils  located  at  right  angles,  and  a  movable  iron  core  which  formed 
part  of  the  magnetic  circuit  and  permitted  more  or  less  of  the  primary 
flux  to  pass  through  the  secondary.  This  device  is  sometimes  referred  to 
as  a  "  magnetic  shunt." 

For  a  description  of  different  forms  of  regulators,  see:  "Alternating 
Current  Feeder  Regulators,"  by  W.  S.  Moody  (a  paper  before  the  Toronto 
Section,  A.  I.  E.  E.,  February,  1908)  ;  "  Alternating  Current  Potential 
Regulators,"  by  G.  R.  Metcalfe,  Electric  Journal,  August,  1908. 


7-B] 


INDUCTION   REGULATORS. 


253 


simultaneously.)  Plot  a  curve,  as  Fig.  i,  showing  the  secondary 
potential  for  various  angular  positions  of  the  rotor.  Note  the 
ratio  of  maximum  secondary  to  primary  potential. 

§7.  (b)  Use  as  Auto-transformer. — The  second  and  commer- 
cially preferred  method  for  use  as  a  single-phase  regulator  is  to 
supply  the  primary  with  single-phase  constant  voltage  as  before, 
and  to  connect  the  secondary  in  series  with  the  load,  as  in  Fig.  2, 


POSITION  OF  ROTOR 

FIG.  2.     Connections.  FIG.  3.     Delivered  voltage. 

Single-phase   potential    regulator     used    as    auto-transformer. 

so  that  the  delivered  potential  E  taken  from  the  machine,  now 
acting  as  an  auto-transformer,  is  equal  to  the  primary  potential 
Elt  either  increased  or  decreased  by  the  potential  E2  of  the  sec- 
ondary. This  is  either  additive  or  subtractive,  the  apparatus 
being  a  booster  or  a  negative  booster,  according  to  the  position 
of  the  rotor.  In  this  manner,  the  potential  may  be  varied  be- 
tween the  limits  of  Et  +  E2  and  Et  —  E2.  Measure  Elt  E2  and 
E.  Plot  a  curve,  as  in  Fig.  3,  showing  the  delivered  potential 
for  various  positions  of  the  rotor.  Compare  E  with  the  algebraic 
sum  of  EI  and  E2.  What  relation  is  there  between  the  curves 
in  Figs,  i  and  3? 

§  8.  A  Comparison. — In  method  (a),  the  output  of  the  regu- 
lator is  equal  to  the  volt-ampere  capacity  of  the  secondary;  in 
method  (&),  the  output  of  the  same  regulator  is  much  greater. 
Take,  for  example,  a  regulator  with  primary  for  100  volts  X  100 


254  PHASE   CHANGERS,   ETC.  [Exp. 

amperes,  and  secondary  for  10  volts  X  1000  amperes.  In  method 
(a),  the  secondary  output  is  limited  to  10  volts  X  1000  amperes, 
or  10  kilowatts.  In  method  (&),  the  potential  may  be  varied 
from  90  to  no  volts,  which  with  1000  amperes  gives  an  output 
of  about  100  kilowatts.  In  practice,  method  (b)  is  therefore 
used;  in  the  laboratory,  method  (a)  is  often  convenient  when  the 
range  of  delivered  voltage  desired  does  not  exceed  E2. 

§9.  Again,  to  take  care  of  a  certain  load  as  a  transformer,, 
the  regulator  must  have  a  capacity  (E2I2)  equal  to  the  load,  72 
being  load  current  and  E2  being  load  voltage.  As  an  auto-trans- 
former, the  regulator  will  have  a  capacity  (E2I2)  which  is  much 
smaller,  72  being  load  current  and  E2  the  increase  or  decrease 
of  voltage  (see  Fig.  2)  ;  thus,  if  the  voltage  is  to  be  raised  or 
lowered  10  per  cent.,  the  capacity  of  the  regulator  needs  to  be  only 
10  per  cent,  of  the  full  load  of  the  feeder. 

§  10.  Further  Experiments. — The  regulator  may  be  tested  under  load, 
either  inductive  or  non-inductive,  as  in  Exp.  5-A;  or  its  performance  can 
be  predetermined,  as  in  Exps.  5-B  and  5-C. 

The  air  gap  necessitates  a  larger  magnetizing  current  than  in  a  trans- 
former with  a  closed  magnetic  circuit;  and,  on  account  of  larger  leakage 
reactance,  gives  poorer  regulation  and  a  smaller  diameter  to  the  circle 
diagram. 

§  ii.  Tertiary  Coil. — As  the  secondary  coil  moves  away  from  the  influence 
of  the  primary  and  comes  more  nearly  to  the  neutral  position,  it  includes 
less  and  less  of  the  primary  flux;  the  secondary  leakage  flux  now  causes 
the  secondary  to  act  more  and  more  as  a  choke-coil  in  series  with  the  load, 
thus  giving  a  low  power  factor.  This  has  been  overcome  by  a  short- 
circuited  tertiary  coil,  wound  midway  between  the  primary  windings,  so 
located  that  it  is  not  cut  by  the  primary  flux.  As  the  secondary  moves 
away  from  the  influence  of  the  primary,  the  short-circuited  winding  comes 
into  play,  acting  similarly  to  a  short-circuited  secondary  on  a  transformer, 
so  that  the  choking  effect  of  the  secondary  of  series  coils  becomes  less  and 
less,  and  is  practically  zero  in  the  neutral  position.  (See  citations,  §4a; 
also  Standard  Handbook,  6-158.)  In  a  polyphase  induction  regulator  no 
tertiary  coil  is  needed  (Standard  Handbook,  6-161). 

§  12.  (2)  Induction  Regulator  as  Phase-shifter.— Supply  the 
primary  with  polyphase  current  at  normal  constant  voltage.  The 
secondary  voltage  will  be  found  to  be  constant  in  value  for  all 


7_B]  INDUCTION   REGULATORS.  255 

positions  of  the  rotor  (instead  of  varying  as  in  the  preceding 
tests),  but  to  be  of  varying  phase,  having  a  definite  phase  posi- 
tion for  each  position  of  the  rotor.  This  should  be  explained  and 
demonstrated  experimentally. 

To  do  this,  connect  one  primary  circuit  and  one  secondary  cir- 
cuit in  series  as  before,  measure  £±  and  E2  separately,  and  E  the 
sum  of  the  two  for  each  position  of  the  rotor,  and  construct 
triangles  on  a  common  base  Elt  as  in  Fig.  4,  which  illustrates  the 
varying  phase*  of  E2.  Observe 
the  relation  between  mechanical 
and  electrical  degrees,  noting, 
for  example,  the  mechanical 
angle  through  which  the  rotor  is 

FIG.  4.     Voltage  relations  as  phase 

turned  to  shift  the  phase  of  E2  shifter, 

by  45  electrical  degrees. 

Although  of  little  commercial  use,  this  method  is  extremely 
useful  in  the  laboratory.  If  the  primary  supply  is  symmetrical 
and  of  constant  voltage,  the  secondary  voltage  on  open  circuit 
will  be  constant  and  its  phase  angle  will  vary  exactly  with  the 
position-angle  of  the  rotor,  which  can  be  read  with  a  suitable 
scale.  A  secondary  load  will,  however,  distort  these  conditions, 
so  that  the  scale  reading  will  not  give  the  phase  exactly. 

The  varying  resultant  potential  E,  in  Fig.  4,  shows  that  with 
polyphase  supply  the  apparatus  can  also  be  used  as  a  potential 
regulator,  to  be  discussed  in  the  next  paragraph. 

§  13.  (3)  Polyphase  Potential  Regulator. — The  primary  is 
supplied,  as  in  (2),  with  polyphase  current  at  normal  constant 
voltage.  The  secondary  coils  for  each  phase  must  be  separate 
from  each  other,  one  secondary  coil  being  connected  in  series 
with  each  delivery  circuit.  For  a  3-phase  regulator  (or  3-phase 
motor  used  as  a  regulator)  the  connections  are  shown  in  Fig.  5. 
The  supply  circuit  is  connected  to  the  terminals  I,  2,  3  of  the 
primary — which  may  be  star-connected,  or  delta-connected  as 

*  This  can  also  be  shown  by  a  phase-meter. 


256 


PHASE    CHANGERS,    ETC. 


[Exp. 


shown.  The  delivered  currents  are  taken  from  a,  b,  c.  The 
voltage  relations  are  shown  in  Fig.  6,  where  a',  b',.c'  gives  the 
maximum  delivered  voltage.  As  the  rotor  is  turned,  this  becomes 
a",  b" ',  c" ,  etc.,  until  the  minimum  a"',  b"' ,  c'"  is  obtained. 


FIG.  5.     Connections. 


FIG.  6.     Voltage  relations. 


Polyphase  potential  regulator:  supply  voltage,  i,  2,  3;  secondary  coils,  x,  y,  s, 
in  series   with  load ;    delivered  voltage,  abc. 

With  a  voltmeter,  show  that  the  secondary  voltages  x,  y  and  z 
do  not  change  in  value  with  a  change  in  position  of  the  rotor; 
also  show  that  the  three  delivered  voltages,  ab,  be,  ca,  are  sub- 
stantially equal  for  any  one  position  of  the  rotor.  (That  x,  y,  z 
do  not  change  and  ab,  be,  ca  change  simultaneously,  as  the  rotor 
changes,  is  well  shown  by  incandescent  lamps.) 

Measure  one  delivered  voltage,  as  ab,  for  different  rotor  posi- 
tions, noting  particularly  the  positions  for  maximum  and  mini- 
mum values,  and  plot  a  curve,  as  in  Fig;.  3. 

Construct  a  diagram  to  scale,  as  in  Fig.  6,  making  the  triangle 
I,  2,  3  equal  to  the  primary  voltage;  the  circles  have  radii  equal 
to  the  secondary  voltages,  x,  y  and  s.  From  this  diagram  pick 
off  values  of  delivered  voltage,  a'b',  a"b" ,  etc.,  for  different  rotor 
positions  and  plot  these  values  as  a  second  curve,  to  be  compared 
with  the  first  curve  already  plotted  from  measurement.  The 
limiting  values  of  delivered  voltage  are  shown  to  be  Ei  ±  2E^ 
cos  30°. 


INDEX. 


Acyclic  dynamo,  2 

Adams,    C.    A.,    on    polyphase    power 
measurements,    230 

Admittance,  104,  115 

Aging  of  transformer  iron,   174 

Alternators,   armature  reaction  of,  94 
auxiliary  field  winding,  69 
characteristics  of,  62-72 
constant  current,  67,  88 
components   of  magnetic  flux  in, 

74 

constant  potential,  67,  88 
composite  winding,  69 
design     as  affected  by  the  steam 

turbine,  62 

determining  efficiency  of,  71 
electromotive    force    method    for 

predetermining     characteristics 

of,   75,  80-90 
impedance   ratio   of,  80 
Institute    rule    for    regulation    of, 

93 

magnetomotive  force  method   for 
predetermining     characteristics 

of,  75,  Qi 
predetermining  characteristics  of, 

73-101 

regulation  of,   66,  69 
regulation  at  lower  factors,  99 
split-field   method   of    testing,    98 
synchronous  impedance  of,  79 
synchronous  reactance  of,  80 
tests  at  low  power  factors,  98 
types    defined,    63 
variation    of   characteristics   with 

power  factor,  70 
Ammeter,  correction  when  used  with  a 

wattmeter,    148 

current  transformers  for,   149 
methods      of      connecting      when 

used  with  a  wattmeter,   148 
Ampere-turn  method  of  testing  alter- 
nators,   75 

Apparent  power,  113 
Arakawa,    B.,    on    vector    representa- 
tion    of    non-harmonic    alternating 
currents,   182 

Armature     characteristic,     of     A.     C. 
generators,    69 


Armature     characteristic,     of     D.     C. 

generators,  23 

Armature  insulation,  drying  by  short- 
circuit  current,  78 

Armature  reactions,  demagnetizing 
and  cross-magnetizing  effect 
of,  6 

effect  on  alternator  regulation,  73 
effect  on  brush  position,  6 
effect   on    series   characteristic,   9 
in  alternators,  74,  94 
in  D.  C.  generators,  19 
in   motors,   31 
local    self-induction    of    armature 

conductors,   19 

Armatures,  closed  coil,  open  coil,  lap 
or    parallel    winding,    wave    or 
series  winding,  3* 
function  of,  in  generators,  i 
peripheral  speed  of,  6 
resistance  of,   12,  42,  76 
Asynchronous  machine,  62 
Auto-transformers,   134-136 
advantages  of,  135 
as  boosters,   135 
induction   regulators   as,   253 
phase    relation    of    primary    and 

secondary   currents,    135 
step-up  and  step-down,  135,  136 
Auxiliary  field  winding,  69 
Average  value  of  a  sine  wave,  146 

Balance  coil,  135 

Balanced  load,    197 

Bedell,     F.,     on     separation     of    iron 

losses,  176 
on    transformer    regulation,    167, 

*93 

Bedell  and  Crehore,  on  equivalent  re- 
sistance and  inductance,   119 
on    effective   and    average   values 

of  a  sine  wave,  146 
on  current  locus  when  resistance 
is  varied   in   an   inductive   cir- 
cuit,  123 
on     three-voltmeter     method     of 

measuring  power,    118 
Bedell    and    Tuttle,    on    the    effect    of 
iron    in    distorting   alternating   cur- 
rent wave  form,  182 


18 


257 


258 


INDEX. 


Bedell   and   Steinmetz,   on   reactance, 

n5 

Belt  losses,  55 
Berg,   E.  J.,  harmonics  in  alternating 

currents,  217 
Brush  position,  in  generators,  6,  9 

in  motors,  31 
Behrend,  B.  A.,  alternator  regulation, 

75,  97,   100 

split  field  method  of  testing,  98 
Blondel,  A.,  alternator  regulation,  100 
loading  back  method,  56 
on  polyphase  power,  240 
Burt,  A.,  polyphase  power  factor,  223 

Capacity,  circuits  with,  120,  121 
Characteristics  of  alternators,  62-101 

armature,  65,  69 

electromotive    force    method    for 
determining,  73-96 

external,  65,  67,  88 

full-load  saturation  curve,  65-67, 
77,  89 

magnetomotive  force  method  for 
determining,  73-96 

no-load  saturation  curve,  65,  66, 
77,  89 

predetermination  of,  73-101 
Characteristics    of   compound   genera- 
tors, 13-26 

armature,  17,  23 

compound,  20,  21 

differential,   17,  21,  22 

full-load  saturation   curve,   26 

no-load  saturation  curve,  14 

series,  17,  21,  23 

shunt,  17,  1 8,  21 

Characteristics     of     D.     C.     motors, 
27-40 

compound,  37,  38 

differential,   37,  38 

series,  39,  40 

shunt,   37,   38 

Characteristics    of    series    generators, 
5-i2 

external  series,  7,  8,  9 

magnetization  curve,  5,  9 

total  series,  8,  9 

Characteristics    of    shunt    generators, 
14—20 

armature,  17,  23 

external  shunt,   17,  18 

full-load  saturation   curve,   26 

no-load   saturation   curve,    14 

total  shunt,  18,   19 


Circle  diagram,  for  circuits  with  re- 
sistance and  reactance,  123- 
127 

for  constant  potential  transform- 
ers,   179-195 
Circular  mils,  5 
Closed  coil  armature,  3 
Close  regulation,  66 
Coefficient  of  self  induction,  103 
Coil  voltages,  3 
Coils,   polarity    tests    of    transformer, 

in,   132-134,   142-143 
Commutator,  3 

Commutation,   line   or   diameter  of,   6 
Commutating  poles,  33 
Compensated  winding,  33 
Compensator,  135 
Composite  field,  69 
Composite  transmission,  245 
Compound  generators,   13-26 

characteristics  of,  see  Character- 
istics 

efficiency  of,  58 

uses  of,  4 
Compound  motors,  27—61 

characteristics  of,  see  Character- 
istics 
Conductance,  115 

of  parallel  circuits,  120 
Constant  losses,  47 
Converter,   synchronous,   62 
Copper  losses,   in   alternators,   72 

in  generators  and  motors,  45-49, 

57 
in    transformers,    129,    151,    161, 

165,  220-221 
Core  losses,  in  transformers,  129,  151, 

155-158,   173,   174 
Cosine   method,    of    measuring   power 

factor,  225 

Cotterill,  belt  losses,  55 
Counter-electromotive      force,      in      a 

motor,  28-31,  34,  50 
Cross-magnetizing    flux,     in     alterna- 
tors, 74,  95 
in   generators,   6 

Current,  apparatus  for  obtaining  con- 
stant, 126 

in  a  D.  C.  circuit,  103 
in  an  A.  C.  circuit,  103,  115,  123 
methods  of  adjustment,  10 
per  phase,  212 
Currents,      addition      of      alternating, 

2OI-2O2 


INDEX. 


259 


Current  locus,  when  resistance  is 
varied  in  an  inductice  circuit,  123, 
125-126,  187,  190-192 

Cycle,  62 

Delta  connection,  197,  206-208 

Delta  voltage,  204  * 

Diameter  of  commutation,  6 

Diametrical    6-phase    connection,    211 

Differential  generator,  characteristics 
of,  22 

Differential    motor,    equivalent    shunt 

excitation,  59 
speed  and  torque  of,  36 
tests  of,  59 
uses  of,  4 

Direction  of  rotation,  effect  on  pick- 
ing up  of  shunt  machine,  15 

Double  delta  connection  of  6-phase 
circuits,  211 

Double  T  connection  of  6-phase  cir- 
cuits, 211 

Double  transformation,  244 

Drying  armature  insulation  by  short- 
circuit  current,  78 

Dynamo,  see  Generator 

Eddy  current  loss,  in  generators  and 

motors,   46,   50-53 
in   transformers,    173-176 
Effective  value   of   a  sine   wave   105, 

146 
Efficiency  of  alternators,  71 

of     generators  and  motors,  41-61 
of  transformers,  166,  178 
Electromotive    force,    how    generated, 

i,  2,  28 
method  of  alternator  testing,  75, 

80-90 

non-sinusoidal,  121,  122 
of  self  induction,   107 
Electromotive   forces,   addition   of,  in 

a  series  circuit,  109,  114 
addition  of,  in  polyphase  circuits, 

199-201,  213—214 
Electrical  degree,   105 
Equivalent  inductance,    119 
Equivalent  reactance,   150 
Equivalent  resistance,    119,   150,   159- 

160 

Equivalent  sine  wave,  105 
Equivalent      single-phase      quantities, 

211-213 

Everest,  A.  R.,  on  transformer  regu- 
lation, 193 

Excitation  characteristic,  see  Char- 
acteristics 


External  characteristics,  see  Char- 
acteristics 

External  revolving  field,  63 

External  series  characteristic,  see 
Characteristics 

External  shunt  characteristics,  see 
Characteristics 

Faraday's  disk  dynamo,  2 

Faraday's  principle,   i,   107,   133,  144, 

181 

Field  .compounding   curve,    see   Char- 
acteristics (armature) 
definition  of,  23 
Field  magnets,  i,  4 
Flather,  dynamometers  and  the  meas- 
urement of  power,  41 
Flux,  magnetic  2 

relation    to    electromotive    force, 

144 

unit  of,  1 6 
Flux  density,  see  Saturation 

in  transformers,   139,   158 
Form  factor,  definition  of,  146 
effect  on  iron  losses,   176 
Franklin  and  Esty,  on  acyclic  homo- 
polar    dynamos,    3 
on   maximum    efficiency,    57 
on       predetermining       alternator 

characteristics,    91 
on  regulation  of  alternators,   100 
Frequency,  best  frequency  for  differ- 
ent machines,   64 
effect  on  exciting  current,  153 
effect  on  transformer  losses,   156 
relation  to  speed,  62,  65 
Friction  and  windage,  law  of,   47 

in  series  motors,  61 
Full-load  saturation  curve,  see  Char- 
acteristics 
definition  of,  26 

Gauss,  1 6 

Generators,    alternating    current,    see 

Alternators 

armature  reactions  in,   6,   19 
asynchronous     and     synchronous, 

62 

brush  position,   6 
characteristics  of,  see  Character- 
istics 

compounding  by  added  turns,   26 
compounding   by    armature    char- 
acteristic,   see    Characteristics 
constant   current,    127 
direct  current,   1-26 
efficiency  of,  see  Efficiency 


260 


INDEX. 


Generators,  fields  of,  i,  4 

loading  back,  55 

number  of  poles  in,  27 

study  of,  1-5 

tests  on  polyphase,  70 

torque  in  27,  28 
Gilbert,   16 
Guilbert,  on  alternator  regulation,  100 

Harmonics  in  delta  and  star  connec- 
tions,  217—219 
Hedgehog  transformer,   130 
Henderson    and    Nicholson,    on    regu- 
lation of  alternators,  100 
Henry,  unit  of  self  induction,  108 
Herdt,  L.  A.,  on  regulation  of  alter- 
nators, 100 
Hobart  and   Punga,   on  regulation  of 

alternators,    100 
Homopolar  dynamo,  2 
Hopkinson,  on  motor  testing,   56 
Housman,    R.    H.,    on    separation    of 

losses  in  a  motor,  50 
Hutchinson,   on   motor  testing,    56 
Hysteresis,  coefficient  of,  174 
effect  of  aging  on,  174 
separation  from  eddy  current  loss 

in  motors,  50-53 
separation     from     eddy     current 

loss  in  transformers,   175 
variation  with  speed  and  flux,  46 
Hysteretic  angle  of  advance,  182 

Impedance,  103,  115 
Impedance  ratio,  80,  165 
Impedance  triangle,  109 
Impedance  voltage,   162 
Inductance,    103-121 

calculation  by  impedance  method, 

116 
calculation      by      three-voltmeter 

method,   117 
calculation  by  wattmeter  method, 

116 

Induction   regulators,   250-256 
Inductor  generator,  i,  62,  63 
Induction   motor,  best  frequency   for, 

64 

slip  in,  62 
as  potential   regulator  and  phase 

shifter,   252 

Insulation  tests,  of  transformers,   177 
Internal  revolving  field  alternators,  63 
Internal  shunt  characteristic,  see  Char- 
acteristics 
Interpole  motor,  33 
Iron,  aging  of,  in  transformers,  130 


Kapp,  G.,  on  motor  testing,  50,  56 
Karapetoff,     Vv     on     heat     runs     of 

transformers,    178 
on  motor  testing,  56 
on    alternator    regulation,    100 
Kelley  and  Spoehrer,  on  variation  of 

transformer  core  loss,   175 
Kennelly,   A.   E.,    on   temperature   co- 
efficient of  copper,   ii 
Kirchhoff's  law,  200 

Lap  winding,  3 

Line  current  of  3-phase  system,  205 

Line  drop  in  polyphase  system,  202— 

204 

Line  of  commutation,  6 
Line  voltage  of  3-phase  system,  204 
Lloyd,  M.  G.,  on  iron  losses,  176 
Load  losses,  defined,  46,  72 

in  transformers,  161 

Institute   rule   for   estimating,    72 
Loading  back,  a  generator,  55 

a  transformer,   177 
Local  armature  reaction,  74 
Long  shunt,   compound  generator,   21 

Magnetic  flux,  2,  16 

Magnetic  leakage  in  transformers,  137 

Magnetic    shunt,    potential    regulator, 

252 

Magnetic  units,   16 

Magnetization    curve,    see    Character- 
istics 

Magneto-generators,  4 
Magnetomotive    force    method    of    al- 
ternator testing,   75,  91 
Maximum  efficiency,  in  motors,  49,  56 

in  transformers,  167 
Maxwell,   16 

McAllister,  A.  S.;  on  changing  from 
3-phase  to  6-phase  converters, 
211 

on   equivalent  single-phase  quan- 
tities, 212 

on  measuring  torque,  40 
on    power    factor    of    unbalanced 

systems,   223 
on   2-  to  6-phase  transformation, 

248 

Mesh-connected  systems,  196,  197,211 
Mesh    method    of    representing    alter- 
nating   currents    and    electromotive 
forces,  216 

Metcalfe,    G.    R.,    on    alternating   cur- 
rent potential  regulators,  252 
Monocyclic   transformation,   246 


INDEX. 


261 


Moody,  W.  S.,  on  alternating  current 

feeder  regulators,  252 
Mordey,  W.  M.,  on  divided  armature 
method  of  alternator  testing,  98 

on  separation  of  losses  in  motors, 

So 
Motors,  armature  reactions  in,  31 

asynchronous,   62 

best   frequency   for   induction,   64 

best   frequency   for  series,   64 

brush  position  in,  31 

compound  wound,  4 

copper  losses  in,  46 

damage  of,  by  field  discharge,  35 

differential  wound,  4 

effect  of  breaking  field  of,  when 
running,   34 

efficiency  of,  by  the  measurement 
of  losses,   41-61 

interpole,   33 

iron  losses  in,  46 

operation   and   speed   characteris- 
tics of,  27-40 

rotation  losses  in,  analyzed,  50 

shunt  wound,  4 

Reparation  of  .losses  in,  51-54 

series  wound,  4 

speed  control  of,  32 

speed  equation  of,  30 

speed  of  shunt,   31 

speed  regulation  of,  37 

stopping  of,   34 

synchronous,  62 

torque  in,  28,  30 

motor  generator,  efficiency  of,  54 
Motor  starters,   automatic   release,   34 

multiple  switch,  34 
Multipliers,   for  ammeters,   149 

for  voltmeters,  149 

for  wattmeters,   149 

Neutral  position   of  brushes,  6 

Noeggerath,  acyclic  homopolar  dy- 
namos, 3 

No-load  saturation  curve,  see  Char- 
acteristics 

Non-inductive  circuit,    103 

Notation    for   polyphase   circuits,    215 

Oersted,  unit  defined,   16 
Open  coil  armature,  3 
Open    delta    in    3-phase    system,    197, 
209 

Parallel  winding,  3 

Peripheral   speed,   of  armatures,  6 


Polygon    method    of    representing    al- 
ternating currents,  207,  216 
Percentage  of  saturation,   16 
Phase    shifters,    induction    regulators 

as,  254 

Polyphase  currents,   196-221,  222-240 
Polyphase  generators,   70 
Polyphase  systems,   196-241 

addition  of  currents  in,  201 
addition    of   electromotive   forces 

in,  199 

copper  economy  of,  220 
current  and  voltage  per  phase  in, 

212 

equivalent     single-phase    quanti- 
ties  in,   211 
harmonics  in,  217 
line  drop   in,  202 
mesh     method     of     representing 

A.  C.  quantities  in,  216 
methods  of  connecting,   196 
polygon    method    of    representing 

A.   C.  quantities  in,   207 
power  factor  of,  223 
radial     method     of     representing 

A.   C.  quantities  in,  216 
6-phase    systems,    210 
uniform  flow  of  power  in,  241 
Polyphase   transformation,    241-249 
Polyphase  wattmeter,  229 
Porter,    on    radial    method    of    repre- 
senting  alternating   current   quanti- 
ties, 216 
Potential   regulators,    250-256 

use  of  induction  motors  as,  252 
induction   regulators,   251 
magnetic    shunt,    252 
polyphase,  255 
single-phase,   252 
step-by-step  type,  251 
use   as  transformers,   252 
Potier,  on  motor  testing,   56 
Power,   definition    of,    116 

in  a  non-inductive  resistance,  113 
in  an  inductive  circuit,  113 
measurement     by     three-ammeter 

method,    118 
measurement    by    three-voltmeter 

method,   118 
measurement     of,     in     polyphase 

systems,    222—240 
n   wattmeter   method   of   measur- 
ing, 228 

n-i    wattmeter   method    of   meas- 
uring,  226 

one   wattmeter   method    of   meas- 
uring 3-phase  power,  233 


262 


INDEX. 


Power,   proof   of   wattmeter   methods 

of  measuring  power,  240 
three-phase,    205,    206 
two    wattmeter    method    for    any 

3-wire   system,  228 
Power  current,   115 
Power  factor,   103,   104,   114 
by  sine  method,  237 
by  tangent  method,  237 
for  equivalent  sine  waves,  105 
for  non-sine  waves,  226 
polyphase,  223,   225,   232,   237 
Predetermination,  of  alternator  char- 
acteristics,  73-101 

Puffer,   testing   of  motors   by   opposi- 
tion method,  56 

Quadrature    relation    of    current    and 

electromotive  force,  106 
Quarter-phase  2-phase  system,  196 

Radial  method  of  representing  alter- 
nating   currents    and    electromotive 
forces,  216 
Reactance,  103,  115 

effect  of  frequency  on,    121,   122 

by  impedance  method,  116 

of  a  circuit  containing  inductance 

and   capacity,    121 
of  a  series  circuit,  109 
by  three-voltmeter  method,   117 
by  wattmeter  method,   116 
Reactance    drop,     in    polyphase    sys- 
tems, 203 

in  transformers,   129 
in   alternators,   67,    74,   80-96 
Reactance  method  of  testing  alterna- 
tors,      see       Electromotive       force 
method 

Reactive    factor,    114 
Regulation,   Institute   rule   on   compu- 
tation of,  93 
of  alternators,  66,  67,  69,  71,  73, 

82-84,  87,  99-100 
of  generators,   19,  22 
of  motors,   32 
of  transformers,  167,  193 
of  transmission  lines,  101 
Reluctance,  2 
Residual  magnetism,  effect  on  picking 

up  of  series  machine,  8 
effect  on  picking  up  of  shunt  ma- 
chine,  15 

Resistance,  109,  115 
Resistance   drop,   in  alternators,  "66 

73,  76,  79,  80,  81-83,  86 
in  generators,  8,  19,  24 


Resistance  drop,  in  motors,  29,  31 
in   polyphase   systems,   202 
in    transformers,    129,    162,    165, 

168-169,    185 
Resistance   measurements,    by   fall-of- 

potential  method,  n 
by  substitution,   12 
Resonance,  current,   121 

voltage,   121 

Revolving  armature,  2,  63 
Revolving  field,  2,  63 
Ring  connected  2-phase  system,   196 
Robinson,    L.    T.,    on    electric    meas- 
urements   with    current    and   poten- 
tial  transformers,    149 
Roessler,  on  form  factor,  146 

on    influence    of    form    factor   on 

iron   losses,    176 
Rotary  converter,  best  frequency  for, 

64 

Rotation  losses,  47,  50,  60 
Rushmore,    D.    B.,    on    regulation    of 

alternators,  91,  100 
on  revolving  field  alternators,  63 
Ryan,    H.    J.,   on    compensated   wind- 
ings, 33 

Saturation,  effect  of,  on  compound- 
ing, 13,  14 

effect  of,  on  regulation  of  com- 
pound generator,  22 

effect  of,  on  regulation  of  shunt 
generator,  20 

effect  of,  on  series  characteristic, 

7 
effect    of,     on     speed     of     series 

motor,  40 

Saturation  curve,  14 
Saturation  factor,  16 
Scott,  C.  P.,  on  two-  to  three-phase 

transformation,   243 
Separation    of    losses,    in    generators 

and  motors,  50-54 
in   transformers,    175-176 
Series  A.  C.  circuits,  102-122 
Series   A.    C.   motors,   best   frequency 

for,  64 
Series     characteristic     of     compound 

generators,  see  Characteristics 
Series  generators,   1-12 
armature  reactions,   6 
brush  position,  6 
characteristics  of,  see  Character- 
istics 

uses  of,   4 

Series  motors,  4,  39-40,  60-61 
Series  turns,  determination  of,  24-26 


INDEX. 


263 


Short  shunt  connected  generator,  21 
Shunt  generators,   13-20 

armature  reactions  in,  19 

characteristics  of,  see  Character- 
istics 

compounding  of,  23-24,  26 

direction  of  rotation  of,  15 

efficiency  of,   41-59 

regulation  of,  19,  20 

uses  of,  4,  13 
Shunt  motors,   27-61 

armature  reactions  in,  31-32 

brush  position  of,  31-32 

efficiency  of,  41-46 

operation  of,   27-40 

speed  characteristics  of,  37~4<> 

speed  control  of,  32-34 

stopping  of,  34 

Shunt  turns,  determination  of,  25 
Sine  method,  of  measuring  power  fac- 
tor, 226,  237 

Single-phase  currents,   102-122 
Six-phase  circuits,   210-211 
Smith,    S.    P.,    on    alternator    regula- 
tion, 99 

Space  degrees,   106 
Sparking  at  brushes,  6,   22 
Speed  control  of  motors,  32 
Speed  equation  of  motors,  30 
Speed  regulation,  37. 
Speed,  relation  to  frequency,  65 
Split-field  method   of   alternator  test- 
ing, 98 
Star-connected  2-phase  system,   196 

3-phase  system,   197,   204 
Star  current,  205 
xStar  voltage,  205 
Starting  boxes,  34 
Static  torque,  29 
Steam  turbine,  influence  on  design  of 

alternators,  62 

Steinmetz,    C.    P.,    on    choice   of   fre- 
quency, 64 

on  definition  of  a  balanced  poly- 
phase  system,    197 

on    form    of    external    alternator 
characteristics,    89 

on  hysteresis  exponent,   174 

on      monocyclic     transformation, 
247 

on  separation  of  iron  losses,  176 

on  topographic  method,   198,   199 

on  wave  form,  217 
Steinmetz    and   Bedell,    on    reactance, 

US 
Stray  power,  47 

method  of  motor  testing,  41-61 


Susceptance,  115 

of  parallel  circuits,  120 

Swenson  and  Frankenfield,  on  motor 
testing,  56 

Symmetrical  polyphase  system,   196 

Synchronous  machine,  definition,  62 

Synchronous  generators,   see   Genera- 
tors 

Synchronous   impedance,   79,   80 

Synchronous    reactance,   80 

Synchronous  watt,  29 

T-connected  transformers,  delta  equiv- 
alent of,  249 

for    2-    to    3-phase    transforma- 
tion, 243 

star  equivalent  of,  249 
voltage  and  current  relations,  248 
T-connection  of  3-phase  circuits,  197, 

208 
Tangent  method  of  measuring  power 

factor,  225,  232,  237 
Teaser    circuit     of     monocyclic     sys- 
tem, 247 

Temperature  coefficients  of  copper,  n 
Temperature       corrections,       formula 

for,  10 
Temperature     of     transformers,     see 

Transformers 
Temperature     rise,     computed     from 

change  in  resistance,   n 
Third  harmonic,  in  delta-connections, 

217 

in  generator  coils,  219 
in  star-connections,  218 
Thompson,    S.    P.,    on    regulation    of 

alternators,    100 
Three-phase  systems,  197 

delta  and  star  currents  and  vol- 
tages, 204 
measurement    of    power    in,    228, 

230,  233 
power  in,   205 

transformation  to  2-phase,  243 
Thury  system  of  direct  current  power 

transmission,  221 
Time  degrees,  106 
Tirrell  regulator,  13,  70,  251 
Topographic  method,  199,  203 
Torda-Heymann,     on     regulation     of 

alternators,  100 
Torque,   expressions  for,   28 
how  created,   28 
in  a  generator,  27 
in  a  motor,  30 
in  compound  motor,  35 
in  differential  motor,  36 


264 


INDEX. 


Torque,  in  series  motor,  39,  40 

in     single-phase     and     polyphase 

machinery,   197 
static,   29 

Total   characteristics    (see    Character- 
istics), 7,  8,  19 
Transformer,    adjustment    of    voltage 

in   testing,    172 
aging  of  iron  in,  130 
all-day   efficiency,    167 
auto-transformers,  134-136 
best  frequency  for,  64 
circle   diagram,    179-195 
circulating  current  test,  144 
computation  of  efficiency,   166 
constant   current,    127 
constant    current    from    constant 

potential,    190 
copper  loss,  165 
core  loss,   155 
current  density  in,   140 
current  ratio  of,   143 
design  data,   139-142 
electromotive      force     and     flux, 

144,   146 

efficiency,   139,   178 
equivalent  circuits  of,  186-192 
equivalent    leakage    reactance    of, 

150 

equivalent  primary  quantities,  187 
equivalent  resistance  of,  150,  159, 

160 
exciting    current,    137,    151,    153 

154 

flux  density  in,  140 
form-factor,  effect  of,   176 
general   discussion   of,    179-189 
harmonics  due  to  hysteresis,   182 
heat  runs,  177 
heating  of,  130 
hysteresis  in,  see  Hysteresis 
insulation  tests,   177 
impedance,   163 
impedance  ratio,  165 
impedance  voltage,   162 
instruments  for  testing,  171 
load  losses  in,  161 
loading  back  method,  177 
losses   in,    129 
magnetic  densities  in,   141 
magnetic  leakage   in,    137 
magnetizing  .current  in,    154,   180 
maximum  efficiency  of,  167 
net  and  gross  cross-sections,   140 
normal   current  and  voltage  in,  173 
open   circuit  test,    151 
operation   and   study   of,    128-149 


Transformer,  phase  of  primary  and 
secondary  electromotive  forces 
and  currents,  134 

polarity  of  coils,  132,  133 

polarity   test   by   alternating   cur- 
rent,   142 

polarity    test    by    direct    current, 
143 

polyphase,   131,   210 

potential  ratio  of,   143 

ratio  of  transformation   of,   133 

reactance  drop  in,  129 

reactance   of,    163 

resistance  drop  in,   129 

resistance  of,   163 

regulation  of,   139,   167,   193,   194 

secondary  quantities  in  terms  of 
primary,    187 

separation  of  hysteresis  and  eddy 
current  losses   in,   175 

series,    131 

short-circuit  test,  160 

step-up  and  step-down,   128 

systems  of  polyphase  connections, 
210 

T-connection  of,  243 

test    by    the    method    of    losses, 
150-178 

total  voltage  drop  in,  194 

tub   type,    131 

types    of,    130 

variation      of      core      losses      in, 
156-158,    173-176 

voltage    and   current   transforma- 
tion, 128 

volts  per  turn,  141 

weight  of  copper  and  iron  in,  141 
Transmission  lines,  regulation  of,  101 
Two-phase  system,  196 

laboratory  supply,  201 

transformation  to   3-phase,  243 

transformation   to   6-phase,   248 
Unipolar  dynamo,  2 
V-connection,     of     3-phase     circuits, 
197,   209 

of  auto-transformers  for  starting 

motors,   248 

Vectors,  addition  and  subtraction  of, 
213-216 

direction  of  rotation  of,  105 

for  representing  admittances,  118 

for     representing     currents     and 
electromotive  forces,  105 

for    representing    impedance,    re- 
sistance and  reactance,   109 

for      representing     non-harmonic 
quantities,    122 


INDEX. 


265 


Vectors,  in  magnetomotive  force 
method  of  predetermining  al- 
ternator characteristics,  96 

relative  accuracy  when  applied  to 
inductive  and  capacity  circuits, 
122 

significance  of,  105 

Voltage  adjustment,  10 
Voltage  per  phase,  212 
Voltmeters,  damage  due  to  induced 

electromotive    force,    12 
methods  of  connecting  when  used 

with  a  wattmeter,   148 
multipliers    for,    149 
power  consumed  in,  148 

Wattless  current,  115 
Wattless  power,    107 


Wattmeters,     correction     for     power 

consumed   in   instrument,    151 
errors  in,  146-149 
multipliers  for,  149 
n  wattmeter  method  of  measuring 

power,   228,   240 

n-i    wattmeter   method    of   meas- 
uring power,  226,  240 
negative    reading   of,    in    3-phase 

power  measurements,  233 
one  wattmeter  methods  of  meas- 
uring 3-phase  power,  233 
polyphase,  229 

two   wattmeter  method   of  meas- 
uring power,   228 
Wave  winding,   3 
Woodbridge,  on  converters,  211 
Workman,  motor  testing,  56,  61 

Y-connected,  see  Star-connected 


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